Blair, whose premiership has come under growing criticism, has insisted he will not give a timetable for his departure.
To date he has said only that he would stand down before the next general election, due by 2010 at the latest.
The finance minister, Gordon Brown, widely seen as a political rival in the ruling Labour Party, is expected to succeed him.
The Independent on Sunday newspaper said Blair, in office since 1997, quoted unnamed sources as saying the prime minister had told several ministers in recent days that he would go in the second half of next year.
It quoted one government minister as saying Blair had given "almost half the cabinet" private assurances about a departure date.
Another cabinet member, questioned about whether Blair had informed him he would go next summer, said: "I'm not going to tell you exactly what Tony said but I wouldn't disagree with that."
Blair has come under increasing pressure in recent weeks after a poor performance by his centre-left Labour party in local elections and a series of scandals which have prompted allegations of incompetence and sleaze.
'Zone of time'
But Lord Charles Falconer, a cabinet minister who has known Blair since their schooldays, said he was unaware of any planned resignation date.
Blair (R) and Gordon Brown have
an often tense relationship
Asked on BBC television if he knew of a date and was prepared to share it, he replied: "I don't and I'm not."
Asked later on Sky television if Blair were likely to go in the second half of next year, he replied: "I have no idea."
Two other British Sunday newspapers also cited government sources indicating that Blair would step down in about a year.
The Mail on Sunday newspaper said Blair had told Brown in February that he would step down in July of next year.
The Sunday Times, also citing senior sources, said Blair had offered a "zone of time" in the second half of next year.