An article by the paper's media editor Dan Sabbagh claims that Thompson has entered the final stage of his eight-year leadership of the BBC, and is "psychologically ready" to move on to a new challenge.
The report says that he will leave the job that paid him £779,000 last year at the end of 2012, after the BBC's coverage of the Olympics and Queen's Diamond Jubilee, or he will wait until early 2013 to make the move.
On Monday, the BBC Trust indicated that headhunters Egon Zehnder had been appointed to start putting in place plans for finding a successor to Thompson, who has gained plaudits for his reign at the BBC, despite various high-profile controversies.
Lord Patten, the BBC chairman, said in an interview with The Times that Egon Zehnder had begun preliminary work so that "when the time comes" the BBC had an "intelligent view" of possible successors to Thompson.
He also said at the Oxford Media Convention yesterday that he had never belonged to an organisation where succession plans were handled well, and so he was keen to avoid any "blood on the carpet" in the search for the next BBC director general.
Thompson is thought to have informed the Conservative peer at the time of his appointment to the Trust last year that he intended to step down at some point before 2015, although it is unclear whether they discussed an exact timetable for his departure.
Thompson, 54, took over at the corporation following the resignation of Greg Dyke as director general and BBC chairman Gavyn Davies after the controversy around the corporation in the wake of Lord Hutton's report on the death of government weapons inspector Dr David Kelly.
However, his reign has been dogged by various controversies, including the Sachsgate scandal in 2008 involving lewd messages left by Jonathan Ross and Russell Brand on the voicemail of Andrew Sachs during Brand's Radio 2 show.
Thompson has been criticised over the level of BBC executive and talent pay during his tenure, as well as the negotiations with the government over the BBC's tough new licence fee deal in late 2010. He has also faced industrial action on several occasions over changes to the BBC's final salary pension scheme and cuts to BBC services.
Last November, the National Union of Journalists (NUJ) called for a "vote of no confidence" in Thompson, after blaming him for the "butchery" of the corporation's recent cuts.
However, his tenure as Britain's most powerful television executive has largely been viewed as a success, and Thompson is unlikely to be short of job offers if he does go, both here and in the US, where he could most likely earn a much higher salary.
The Guardian reports that the BBC could look to replace him with an internal candidate, including the potential appointment of the first woman to run the BBC.
BBC chief operator officer Caroline Thomson and head of news Helen Boaden are both considered viable candidates, but they could face competition from George Entwistle, who was appointed the head of TV channel division BBC Vision last year.
Jana Bennett, who Entwistle replaced at BBC Vision, could also be a contender, after she moved to a senior role at commercial arm BBC Worldwide in 2011, overseeing all the corporation's overseas TV channels and the global version of BBC iPlayer.
Outside the BBC, ITV's director of television Peter Fincham could be in with a shout, along with Channel 4 chief executive David Abraham.
The BBC has not yet commented on the report.