Murphy, a former Reds midfielder, rates Hodgson highly after the English boss led the Cottagers from the brink of relegation to last season's Europa League final.
Hodgson has struggled early on at Anfield, suffering defeats to Blackpool in the Premier League and Northampton in the Carling Cup, and the possibility of the club being sold to New England Sports Ventures in the next week has heightened speculation surrounding his future just months into the job.
But Murphy has urged the club to show patience towards their manager.
He told The Independent on Sunday: "The way he wants his team to play and the type of tactics and discipline he wants his team to have don't happen overnight.
"Of course the Blackpool and Northampton results were unacceptable. I'm not hiding behind that. All I'm saying is the situation is not dire. The league table is separated by five points from third-bottom to fourth, so there's a long way to go.
"Stay patient, the team has quality in it and the fact is you panic now and, God forbid, got rid of him and brought someone else in, the problems are still there. Give Roy a bit of time and by Christmas see where we are and everyone will be happier."