The Steelers also made qualifying offers to wide receiver Emmanuel Sanders and nose tackle Steve McLendon. The offers give the Steelers the right to match any offer sheet the players may receive from other teams during free agency.
Dwyer has already signed his sixth-round tender, worth $1.323 million, a source told ESPN NFL Insider Adam Schefter.
Dwyer and Redman likely will battle for the starting running back spot next season. Veteran running back Rashard Mendenhall, a free agent, will almost certainly not return to the Steelers. Dwyer and Redman combined for 1,033 yards rushing and four touchdowns in 2012 but struggled staying healthy.
Sanders will move to the team's No. 2 receiving spot if free agent Mike Wallace leaves.
Information from ESPN NFL Insider Adam Schefter and The Associated Press was used in this report.