The England lock was initially ruled out for 'approximately' 12 weeks after undergoing an ankle operation at the start of October.
Lawes picked up the problem in a game against Sale Sharks at the end of September and looked likely to miss the rest of 2020.
But following the 49-29 defeat to Harlequins at Franklin's Gardens on Saturday, Boyd was able to provide positive news on Lawes.
"Courtney will be available next week (against Bristol) or the week after against Bordeaux - one of the two," Boyd said.
Saints are set to welcome back some key men in the weeks to come, with Mike Haywood recovering well from injury and Dan Biggar only having one more Wales game to negotiate.
Boyd said: "Courtney Lawes has got to come back, Mike Haywood, Dan Biggar - three influential people, but your team's got to be bigger than three.
"We had our England guys back (against Harlequins) and they were a little bit rusty as they haven't played many matches.
"We've got some players to come back but at the end of the day, we've got to be better than that."