1141 GMT March 31, 2020
Pressure has been growing on the Norwegian for some time, BBC wrote.
Some fans believe the defeat by its rival at Hampden Park was the final straw, with a number taking to social media to call for his sacking.
But it is understood that the Glasgow club will now concentrate on retaining the Scottish Premiership title with Deila in charge.
The 40-year-old's side was beaten by the championship winner after a penalty shoot following a 2-2 draw with their city rival.
However, despite the man himself admitting he faces an uncertain future, it would appear that Celtic intends to continue with Deila until at least the end of the season.
It is eight points clear of Aberdeen at the top of the table with five games remaining.
And there is a fear that any change of manager now might disrupt its progress to what would be a fifth consecutive league title.
Deila, which previously led Stromsgodset to the Norwegian title, took over at Celtic after the departure of Neil Lennon in 2014.
He led Celtic to the league title, while the team also lifted the Scottish League Cup, in his first season.
On the downside, it lost to Inverness Caledonian Thistle in last season's Scottish Cup semifinal and to Ross County in this campaign's League Cup last four.
Deila's side also lost to Malmo in a playoff to reach the Champions League group stage before finishing bottom of their Europa League section.