Allardyce's West Ham side beat struggling Sunderland at the weekend, a victory which was only their second league win in 2015.
Speculation over his future has increased over the past month, with SkyBet touting his as the second favourite Premier League boss to be next to get sacked, offering odds of 5/1. Only Leicester's Nigel Pearson (4/1) is considered more vulnerable.
Redknapp is a real fan of Allardyce, but expects manager and club to seek a new direction in the summer.
He said: "I don’t care what anybody says, Big Sam has done a good job at West Ham. He saved them from relegation last year, rebuilt over the summer and some of their attacking football in the first half of the season was fantastic to watch.
"But for whatever reason, his face doesn’t seem to fit and I suspect he will move on in the summer. If that happens, I’m sure there will be plenty of suitors for Sam.
"He is a talented manager and something of a relegation specialist, so don’t be surprised if he waits until a struggling club comes calling."
Despite the poor run West Ham have been on lately, the club are ninth in the Premier League, which shows just how well they started earlier in the season.
Allardyce was asked after West Ham's 1-0 win over Sunderland about recent criticism of results from club vice-chairman Karren Brady, and responded to defend his players and tell SkySports such comments were wrong.
Redknapp had also praised Allardyce after the win via SkySports, citing injuries to key players like Andy Carroll as reason for the club's disappointing form.