England youth star Murphy has just regained full fitness after a problematic foot injury, which he suffered in pre-season.
The 19-year-old Teessider is very highly rated and has already featured 13 times for the first team.
However, injury has beset his progress including a knee ligament problem he sustained at the start of the 2002 season.
Boro boss Steve McClaren is now keen on getting him fully fit and Murphy's agent confirmed that he is looking to get a temporary move.
"Murph wants to get out on loan, just to get some fitness," agent Steven Denos told skysports.com.
"He is local lad and desperate to play again, but he has had really bad luck with injuries.
"A number of clubs are interested and should get something within the next few weeks."
Denos is also the agent of Stockdale, but he is in a different boat to Murphy and is looking for a permanent move away from the club.
The Scottish international is well down the pecking order at Boro and has already spent time with West Ham United this season.
"There are a number of clubs looking at Robbie," Denos continued.
"He knows where he stands at Boro and he is looking for a new start."
Rotherham United and Watford are leading the chase for Stockdale, with the latter also interested in Murphy.
Ipswich Town are also said to be monitoring the pair and both players should feature for Boro's second string next week - when a host of scouts should be present to take in their displays.
