This claim comes from the Guardian, a fairly well-known and reputable source to begin with. This article states that, when referring to the possibility of a third term, Trump "would love to do it." He said that he had his "best numbers ever," indicating his presumption of success in a hypothetical third term. But the 22nd Amendment of the US Constitution bars anyone from running for a third term. In response to this, many have floated the unorthodox idea that Trump run as Vice President under JD Vance's campaign, who would then resign and allow Trump to step up to the Presidential position. In response to inquiries about running for VP, Trump states that he "wouldn't do that," that he "would rule that out," and that "it wouldn't be right." 
NBC News corroborates Trump's stance against running for VP. While NBC did not investigate his quote about "loving" to run for a third term, it did mention that on Wednesday, Oct. 29, Trump admitted that it's "pretty clear" he is not allowed to run for a third term. In supposedly ruling out running for a third term while also claiming that a run for VP wouldn't be right, it seems that Trump will not be vying for President in 2028. https://www.nbcnews.com/politics/donald-trump/trump-rule-out-third-term-constitutional-amendment-rcna240438
Reuters, another reputable news source, corroborates both claims that Trump "would love to" but will not be attempting presidency through the avenue of running for VP, since it "wouldn't be right." Reuters actually follows "Trust Principles," established in 1941, that require all employees to "act at all times with integrity, independence, and freedom from bias." https://www.reuters.com/world/us/trump-says-he-would-not-run-vice-president-2028-2025-10-27/
Again, the New York Times has evidence of Trump's claim to "love to" run for a third term, but that it is "not allowed," and that the approach through running for VP is "ruled out" and "wouldn't be right." https://www.nytimes.com/2025/10/29/us/politics/third-term-limit.html
PBS News also shares Trump's expressed interest in a third term, followed by his claim that while he'd "be allowed" to run for VP, he "wouldn't do that." https://www.pbs.org/newshour/politics/trump-says-its-too-bad-he-cant-run-for-a-third-term
Nearly every article I read similarly establishes claims of Trump loving the idea of a third term, claiming that it could be successful and that it may not be ruled out. Each article subsequently explains the unconstitutionality of a third term, followed by Trump's more recent admissions that a third term is simply not possible. They then go on to quote Trump's claims that he does not support the idea of running for Vice President. 
It seems clear that the claim is true that Trump does love the idea of an unconstitutional third term, but that he will not be pursuing the presidency by running for VP (or by attempting to run at all, it may be reasonably assumed). I have additionally added the YouTube clip where Trump states that he would not run for VP, but that he may love a third term. https://www.google.com/search?sca_esv=bc0aa39c15f4c826&rlz=1C5CHFA_enUS917US917&sxsrf=AE3TifOw2o1EqSg38n44kEvNrihBYvUDXw:1761869957651&udm=39&fbs=AIIjpHxU7SXXniUZfeShr2fp4giZPH5QghoXViUOqdFyhkUfHhA90f6RZnLvcDKUWyhDavEB0GGDOT3-g6-c9W8Pjy1gZAXDnqFJqVXLvD0e_AkTILB1bpX2xQHajXok6NJfp0bhrLs6Y1KmscZmWkoPPVR3ly53S0EAkj0i1284GSNZIyxrDS5syIS7rqefforhRhC9dhLahO8j5Gmjd_vLA6XsTICjPQ&q=trump+%E2%80%98would+love%E2%80%99+unconstitutional+third+term+but+rules+out+running+for+VP&sa=X&ved=2ahUKEwi1jd2Hlc2QAxXeGTQIHY-XChkQs6gLegQIEhAB&biw=1440&bih=778&dpr=2#fpstate=ive&ip=1&vld=cid:57998dd9,vid:Zgawn8_s9Y0,st:0