Martinez had only come back from an ankle injury, which has kept him out for most of this campaign.
Lindelof was also recently returning from absence, but is now set for another spell on the sidelines.
A United statement read: "Manchester United center-backs Victor Lindelof and Lisandro Martinez will be out of action for at least a month due to muscle injuries.
"Sweden captain Lindelof was withdrawn during the second half of last Saturday's Premier League game against Brentford with a hamstring problem. Victor was replaced by Martinez after 69 minutes, as the Argentina international made his first appearance since early February.
Unfortunately, Licha (Martinez) has now sustained a calf strain in training and is unavailable for the next few games."