Gary Bowyer was sacked after the club failed to reach the League Two play-offs.
The 50-year-old was appointed on a short-term basis in March 2021.
He agreed to a full-time role on a two-year deal that May but could only lead the club to a 10th-place finish and a play-off spot.
Billy Barr's contract was terminated by the Ammies.
The managerial position will be announced in due course.
In 2020-21, Bowyer replaced Wellens, and he led them to an eighth-place finish, two points off a play-off place.
Bowyer had previously been under-23s manager at Derby County and was expected to return to the Rams but instead joined Salford on a permanent basis last summer.
Despite a better showing in the second half of the season, Salford missed out on a play-off spot.
Dominic Dietrich is the host of Radio Manchester.
Gary Bowyer has had a measured approach to management, which they lacked last season, having appointed four managers in a single campaign.
The first half of the season was difficult due to injuries to key players like Ian Henderson.
In the January transfer window, they brought in a statement signing from Millwall, Matt Smith, which followed deals for Ryan and Stephen Kelly. The results began to follow once Bowyer had the tools.
The Ammies won nine games and drew six after the window closed, giving them hope of finishing in the top seven.
They lost against rivals in the League Two play-off picture.
With Bowyer gone, whoever comes in will have their own ideas and it could be another summer of change at the Peninsula Stadium.
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