Ryan Giggs is being considered as a future Swansea manager, but the club's owners are keen to give Francesco Guidolin time to turn their season around, according to Sky sources.
Swansea have just one win from five Premier League games and Guidolin is under further pressure following the Swans' 2-1 EFL Cup defeat to Manchester City on Wednesday night.
Chairman Huw Jenkins is a long-time admirer of Giggs, but it is understood the club's new majority shareholders - Americans Jason Levein and Steve Kaplan - are keen to avoid a change so early in the season.
Swansea host City for a second time in four days on Saturday in the Premier League before taking on Liverpool and Arsenal, but Guidolin insists brighter times lie ahead for his side as he plots to arrest their early season slide.
"I am optimistic because I saw my players play a good game," he said.
"At Southampton, we deserved to lose. But in this game we started very well, with focus, aggression, intensity and good ball possession.
"We just have to look forward. If we play well and work hard, I think we are going to have some better moments.
"Saturday against Manchester City will be more difficult than this [EFL Cup tie]. They had a very good team, even if they played without some players, but on Saturday there will be new important players here.
"But we have to prepare a good game and we have to play well, and then we will see.
"The performance of my team for me was good - in the first half and the second half - but sometimes little things go against you, and in this case it was a deflection on the first goal."