Laudrup, who won the League Cup in his first season with Swansea, has steered the club into the FA Cup fifth round, but they are struggling in the Premier League and lie just two points above the relegation zone.
The defeat at Upton Park was their 12th of the season, with the Swans having won just six of their opening 24 games.
Laudrup's troops are 12th in the Premier League table but they are among four teams locked on 24 points and face a fight for top-flight survival.
The Dane has a strained relationship with chairman Huw Jenkins, but remains in the manager's hot-seat with some massive games on the horizon.
However, Sky Sports understands the board are split over his future at the Liberty Stadium.
Swansea face arch rivals Cardiff in their next game, and tackle Everton in the FA Cup and Napoli in the Europa League later this month.