The news comes as a big blow as Boyd's side continue with a hugely difficult and significant run of matches, with a must-win Champions Cup trip to Ulster coming on Friday.
Saints will then face Harlequins, Saracens and Newcastle Falcons in the Gallagher Premiership before returning to Europe for their final two pool games.
And it now appears unlikely that Freeman will be able to play a part in most of, if not all, of those encounters.
"Tommy's probably the one out of last weekend who's not going to make it," Boyd said.
"He's got a sprained shoulder, ACJ, so he's probably four to six weeks away.
"There are probably about 10 who may or may not make it (to Ulster) so we'll have to wait until later in the week to see how that pans out."
But Saints have been able to welcome Courtney Lawes back to training this week after he overcame Covid-19.
"He trained well today so he's back in consideration," Boyd said at Monday's media session.
"We'll manage his load but he hasn't played in the past couple of weeks."
Dan Biggar couldn't complete the game against Ulster as he limped off during the second half.
And Boyd said: "He's having some problems starting and finishing the games.
"We'll assess where he's at with the short turnaround but we're certainly planning to take as strong a team as we can to Ulster."