Garry Monk is not guaranteed to be in charge of Swansea next weekend, according to Sky sources.
Saturday's 3-0 home defeat to Leicester left City two points above the relegation places with just one win in 11 Premier League games.
Monk afterwards admitted he was at a loss to explain the club's "unacceptable" results and told Sky Sports the supporters "deserve better".
It is understood no decision on the manager's future is imminent, but the Swansea hierarchy are reviewing the game against Leicester, and it is not certain that Monk will take the side to Manchester City next Saturday.
The 36-year-old succeeded Michael Laudrup as manager in February 2014 and last season led City to an eighth-placed finish and a record points total.
That achievement helped earn him a new three-year contract in the summer, but results have deteriorated in recent months and Monk has overseen just one win since an unbeaten August.