Albert tore his ACL and MCL in his knee during Week 10 of last season.
The Dolphins will begin training camp on Thursday. Albert will not be full go at the start of camp, according to the source.
But staying off PUP allows him to do limited work and walkthroughs in practice, which can help Albert work toward his stated goal of playing in Week 1.
The Miami Herald first reported the news of Albert's progress.
Dolphins coach Joe Philbin addressed Albert's progress in a one-on-one interview with ESPN.com on Sunday.
"The great news about him is his attitude and the way his mindset is he wants to play football as soon as possible," Philbin said. "So as soon as the medical people deem that's something that he can do, that's great.
"I think we all need to understand there has to be systematic progressions. It's not like we're going to ship him out there Day 1 and have him block [Dolphins defensive end Olivier Vernon] the whole day. We're going to see where he's at, consult with the medical people, and come up with a plan to gradually build him up to where he's ready to play NFL games. How quickly that is? What that looks like? I'm not exactly sure yet."
Albert signed a $47 million contract with the Dolphins in 2014 and was playing at a Pro Bowl level in the first 10 games. Miami's offense suffered significantly after Albert went down.