Lambert retains the faith of Villa owner Randy Lerner even though the Midlands club are also involved in a Premier League relegation battle.
It is believed Lambert would still stay in charge even if Villa drop down to the Championship with Lerner believing he would be the right man to take them back up after his double promotion exploits at Norwich.
Villa suffered a shock 4-3 aggregate reversal against League Two outfit Bradford, who became the first fourth-tier club to reach the final of the competition for 51 years.
Villa now face another tricky tie at Millwall in the FA Cup on Friday before next Tuesday's crucial home Premier League encounter with fellow strugglers Newcastle.
But it is now understood that Lerner remains convinced Lambert is the man to take Villa forward and is aware of the need to keep calm and not overreact.
The American knew it would be a rollercoaster season in what is seen as a long rebuilding job for Lambert after the difficult times under Gerard Houllier and Alex McLeish.
Lerner was happy for Lambert to put faith in young players and the Bradford setback will also not alter Villa's attitude towards the current transfer window.
Lambert has only limited resources available and has said he will not sign players for the sake of it.