It follows Saturday's dramatic 30-27 defeat by Harlequins at Kingsholm, a match they led 27-15 with eight minutes left to play.
Former Brumbies coach Fisher joined the Cherry and Whites in 2014.
A statement from Gloucester said: "The club is aware of the tweet and we are not in a position to comment further."
Gloucester are ninth in the Premiership table after the loss, but have qualified for the European Challenge Cup quarter-finals this season after topping their group with five wins from six.
With five Premiership games remaining, the Cherry and Whites are eight points adrift of the top six and automatic qualification for the European Champions Cup.
Fisher is part of the coaching team that works under director of rugby David Humphreys.
BBC Radio Gloucestershire's Paul Furley
Gloucester's lack of progress up the Premiership table has been a source of frustration for fans for nearly three years since Nigel Davies left in 2014.
Winning the European Challenge Cup in 2015 provided reason for optimism, but crucially did not earn a place at Europe's top table. And now, after losing at home to Harlequins the Cherry and Whites look highly likely to miss out on Europe's top club competition again.
A change in style for the 2016-17 season has enabled Gloucester to turn over Wasps and Saracens at Kingsholm, but Fisher himself admitted this week that they have struggled against teams around them in the table that they need to be beating. As he put it himself: "That obviously says something about our mind set, about our preparation - it's something we're conscious of and are trying to address."
Clearly, having admitted this week's game was a must-win match, Fisher now feels the responsibility to carry the can.
Laurie Fisher's future as head coach at Gloucester is in doubt, after he referred to making "room for someone else" in the role on social media.