League One newcomers Crawley have turned to Barker to replace Sean O'Driscoll, who left the club last month to return to Nottingham Forest without taking charge of a single game.
A deal between the two clubs has yet to be completed, but the matter is expected to be finalised sometime on Tuesday.
A statement on Bury's official club website said: "Bury FC would like to advise all supporters that an official statement is due this afternoon in regards to the future of our manager Richie Barker.
"We have to stress that at this precise moment no agreement has yet been signed but we are expecting to make an announcement this afternoon."
Barker took charge of Bury in April 2011 following Alan Knill's move to Scunthorpe and helped the club to promotion from League Two. The 37-year-old then guided the Shakers to a 14th-place finish in League One last season.
