Red Sox reliever Matt Barnes, on the injured list with right shoulder inflammation, is scheduled to pitch for the Portland Sea Dogs three times this week. Michael Dwyer/Associated Press

BOSTON — Could Matt Barnes finally return from the injured list next week?

Barnes, who has been on the 15-day IL since May 31 with right shoulder inflammation, will make another rehab outing for Double-A Portland on Tuesday in Hartford vs. the Yard Goats, Manager Alex Cora said.

Cora said Barnes then likely will throw in back-to-back games for the Sea Dogs on Friday and Saturday.

The righty has a 7.94 ERA, 5.30 FIP and 1.71 WHIP in 20 outings (17 innings) for Boston this year.

Cora was asked if Barnes might return to the Red Sox next week. He answered, “I believe so. We’ll see.”

The 32-year-old Barnes, who is in the first year of a two-year, $18.75 million contract extension, pitched for Triple-A Worcester vs. Syracuse both Friday and Sunday.

He recorded two outs (both strikeouts) Friday while his fastball was at 94-95 mph. He didn’t allow a run but he gave up two hits and one walk.

He allowed two runs, both earned, on two hits, one walk and one hit by pitch Sunday.

CHRISTIAN ARROYO “most likely” will begin a rehab assignment for Double-A Portland on Wednesday, Cora said.

Rich Hill, meanwhile, will make a rehab start for the Sea Dogs in Hartford against the Yard Goats on Wednesday.

Arroyo, who is on the 10-day injured list with a left groin strain, faced Hill during the lefty’s three-inning simulated game Friday.

Hill is returning from a left knee sprain he suffered during his start in Chicago vs. the Cubs on July 1. Boston placed him on the 15-day injured list July 2.