The Sahih of al-Tirmidhi and the hadith,

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Source of the Allegation

[Nafahat al-Azhar fi Khulasat 'Abaqat al-'Anwar, Sayyid Hamid Husayn Lakhnawi, translated into Arabic by Sayyid 'Ali al-Husayni al-Milani, Qum, vol. 12, p. 124]
 

The Evidence

Al-Tirmidhi lived between the years 209 AH and 279 AH.  Several authors of books of hadith, sirah, history and kalam who came after him acknowledged that he has quoted this hadith - in the exact wording stated above - in his Sahih.  However, the current editions of the Sahih only have the hadith 'I am the house of wisdom and 'Ali is its door'.  This hadith can be seen in the extract below:
 

Sahih (or Jami` or Sunan), al-Tirmidhi (d. 279 AH), p. 141 [Harvard] [London]

Those who say al-Tirmidhi had quoted the 'city of knowledge' hadith

Please note that the following list is only some of those sources of the Ahl al-Sunnah that quote the "city of knowledge" hadith mentioning al-Tirmidhi as one of its narrators.  It does not include the multitude of other sources that narrate this hadith without mentioning al-Tirmidhi's name.
 

Ta'rikh al-Khulafa', Jalal al-Din al-Suyuti (d. 911 AH), p. 170 [Harvard] [London] "... al-Tirmidhi and al-Hakim has quoted on the authority of 'Ali who said: The Messenger of God, peace and blessings be upon him, said: 'I am the city of knowledge and 'Ali is its gate'..."

This hadith is indeed present with the exact stated wording in al-Hakim's Mustadrak `ala al-Sahihayn, vol. 3, pp. 126-7.
 

Al-Durar al-muntatharah fi al-ahadith al-mushtahirah, Jalal al-Din al-Suyuti (d. 911 AH), p. 23 [London] [Harvard] [Ohio state University]

And similarly:
 

Jawahir al-`Iqdayn, Nur al-Din 'Ali al-Samhudi (d. 911 AH), manuscript Al-Tirmidhi has narrated from 'Ali with a raised chain of narration: 'I am the city of knowledge and 'Ali is its gate'.

Al-La'ali al-Masnu`ah, Jalal al-Din al-Suyuti (d. 911 AH), vol. 1, p. 332 [Cambridge] [Oxford] and Al-Nukat al-badi`at `ala al-mawdu`at, Jalal al-Din al-Suyuti (d. 911 AH), manuscript "Hadith 'Taa' 'Kaaf': 'I am the city of knowledge and 'Ali is its gate', it is mentioned from the hadith of 'Ali and Ibn 'Abbas.  I say: The hadith from 'Ali has been narrated from al-Tirmidhi and al-Hakim, whereas the hadith of Ibn 'Abbas has been quoted by al-Hakim and al-Tabarani..."

Nawaqid al-Rawafid, Mirza Makhdum `Abbas b. Mu`in al-Din al-Jurjani (d. 988 AH), manuscript [I.O., Delhi Arabic MS. 972] "The Messenger of God, peace and blessing on him, said: I am the city of knowledge and 'Ali is its gate.  Al-Tirmidhi has narrated it."

Al-Lama`at fi Sharh al-Mishkat, Shaykh `Abd al-Haqq al-Dihlawi (d. 1052 AH), chapter of the merits of  'Ali [California] [Yale] "Know that the famous version of the hadith in this meaning is: 'I am the city of knowledge and 'Ali is its gate'. The critics have spoken about it, and its origin is from Abu al-Salt [al-Harawi] who was a Shi'i and he has been spoken about (i.e. criticised).  This hadith has been considered sahih by al-Hakim, and al-Tirmidhi has considered it hasan..."

Taysir al-Matalib al-Saniyyah (on margin of al-Mawahib al-Ladunniyyah, Nur al-Din al-Shabramalasi (d. unknown), manuscript [while mentioning names of the Prophet i.e. madinatul 'ilm] "... al-Tirmidhi and others have narrated with a raised chain of narration: 'I am the city of knowledge and 'Ali is its gate..."

Al-Nibras li kashf al-'iltibas al-waqi` fi al-'asas, Ibrahim b. Hasan al-Kurdi al-Kawrani al-Shafi`i (d. 1101 AH), manuscript ".. and he was the gate of the city of his [i.e. Prophet's] knowledge according to the saying of the Prophet, peace and blessings upon him: 'I am the city of knowledge and 'Ali is its gate'.  It  was narrated by al-Bazzar and al-Tabarani in al-'Awsat on the authority of Jabir b. 'Abd Allah, and al-Tirmidhi and al-Hakim on the authority of 'Ali."

Sharh al-Mawahib al-Ladunniyyah, Muhammad b. `Abd al-Baqi al-'Azhari al-Zurqani al-Maliki (d. 1122 AH), vol. 3, p. 143 [while explaining names of Prophet i.e. madinat al-'ilm] [Chicago] [California]  [Oxford] "Madinat al-'ilm: as in what the Messenger of God, peace and blessings on him and his progeny, said: 'I am the city of knowledge and 'Ali is its gate'.  It was narrated by al-Tirmidhi and al-Hakim and he considered it sahih as did others [transmitting] on the authority of 'Ali."

Dhakhirat al-Ma'al fi Sharh `Aqd Jawahir al-La'al, Shihab al-Din Ahmad b. `Abd al-Qadir al-`Ujayli al-Shafi`i (d. 13th century AH), manuscript "Al-Tirmidhi narrated that he, peace and blessing on him, said: 'I am the city of knowledge and 'Ali is its gate...'"

Minhaj al-Sunnah, Ibn Taymiyyah al-Harrani (d. 652 AH), vol. 4, p. 138 [London] [Cambridge]

Although Ibn Taymiyyah does not consider the hadith reliable - and his view is thoroughly refuted in Nafahat al-'Azhar (vol. 12, p. 132) - he is still aware that al-Tirmidhi has narrated it in the words of "city of knowledge".  If he believed that this hadith was absent from the Sahih then he would not have lost the opportunity to say so to strengthen the position of his argument.
 
 

Al-Sirah (Subul al-huda wa al-rashad), Shams al-Din Muhammad b. Yusuf al-Salihi al-Shami (d. 942 AH), v. 1, p. 631 [Penn] [London]

The modern editor of Subul al-Huda, Dr. Mustafa `Abd al-Wahid, explains in a footnote that the text of this hadith in al-Tirmidhi is "I am the house of wisdom....".

Ibn al-'Athir can also be seen ascribing the 'city of knowledge' version to al-Tirmidhi:
 

Jami` al-'usul fi ahaditrh al-Rasul, Ibn al-'Athir (d. 606 AH), vol. 9, p.473 [Library of Congress] [London] [Harvard] [Cambridge]

As in the case of Subul al-Huda above, the modern editor of Ibn al-'Athir's book, `Abd al-Qadir al-'Arna'ut, in a footnote to this entry says that the wording of this hadith according to al-Tirmidhi was "I am the house of wisdom (dar al-hikmah) and 'Ali is its door." 

This implies that a leading scholar such as Ibn al-'Athir either considered the two ahadith to be identical, or mistakenly attributed the wrong version to al-Tirmidhi.  It could also mean that al-Tirmidhi quoted this hadith in some other work of his, other than his Sahih (also called Jami`).

A more likely  possibility is that Ibn al-'Athir knew the "city of knowledge" version to be in al-Tirmidhi's Sahih but al-'Arna'ut had to offer the explanation because he could not locate the hadith in it.

Could these authors not have considered the 'city of knowledge' to be the same as the 'house of wisdom' version?

Firstly, the traditionists (muhaddithun) are well-known for their precision when dealing with ahadith with similar but unidentical texts (mutun).  Indeed, the classical scholars considered even identical traditions with different chains of narration to be unique.

Secondly, there is clear evidence that both the 'city of knowledge' and 'house of wisdom' versions were existent in al-Tirmidhi's work.
 

Al-Sawa'iq al-Muhriqa, Ibn Hajar al-Haytami al-Makki (d. 974 AH), p. 122 (p. 73, older editions) [Yale] [London ] "... al-Tirmidhi and al-Hakim on the authority of 'Ali who said: The Messenger of God, peace and blessings be upon him, said: 'I am the city of knowledge and 'Ali is its gate'.... and in another [narration] recorded by al-Tirmidhi on the authority of 'Ali, 'I am the house of wisdom and 'Ali is its door'..."

'Is`af al-Raghibin (margin of Nur al-'Absar), Muhammad b. `Ali al-Sabban al-Misri (d. 1206 AH), p. 156 [London] "... al-Tirmidhi and al-Hakim on the authority of 'Ali who said: The Messenger of God, peace and blessings be upon him, said: 'I am the city of knowledge and 'Ali is its gate'.... and in another [narration] recorded by al-Tirmidhi on the authority of 'Ali, 'I am the house of wisdom and 'Ali is its door'..."

Al-`Iqd al-Nabawi wa al-sirr al-Mustafawi, Shaykh b. `Abdallah al-`Idrus al-Yamani (d. 990 AH), manuscript "Al-Bazzar and al-Tabarani in his al-Awsat have quoted on the authority of Jabir b. 'Abdallah, and al-Tabarani, al-Hakim, and al-'Uqayli in his al-Du'afa, and Ibn 'Adi on the authority of Ibn 'Umar, and al-Tirmidhi and al-Hakim on the authority of 'Ali: The Messenger of God, peace and blessings upon him, said: 'I am the city of knowledge and 'Ali is its gate'.  And in a narration: 'so who desires knowledge should come through the gate'.  In another [narration] from al-Tirmidhi on the authority of 'Ali: 'I am the house of wisdom and 'Ali is its gate'."