The Mirror says Tevez was playing golf in Spain on the day Manchester City produced their worst performance of the season.
The Argentina striker was given permission to take a few days away for family reasons - and missed the 3-1 defeat at Southampton, which leaves City trailing Premier League leaders Manchester United by 12 points.
It is understood that he spent last Saturday on the championship golf course at Valderrama.
But there is no suggestion that Tevez breached any club rules during his leave of absence.
After being regularly photographed playing golf when he went AWOL in Buenos Aires for six months last season, his judgement could be called into question.
A spokesman for Tevez told the Mirror: "Carlos was given permission to take some time off by Manchester City so that he could attend to a personal matter. He was in Spain last weekend and he may have taken a few hours to relax."