Danilo, who recently recovered from a cheekbone fracture, sustained a knee injury in the Gers' 1-0 win over Hearts in the Scottish Premiership on Wednesday, and Clement has since revealed that the striker may require surgery.
Lawrence, meanwhile, was forced off in the same game at Tynecastle with a muscle problem and has been ruled out until after the mid-season winter break.
Speaking at a press conference on Friday ahead of Rangers' home encounter with Dundee on Saturday, Clement said: "Tom will be out for sure until the winter break and Danilo will also be out until that time so that is not so good.
"Danilo has a problem with his knee because of a bad tackle in the game, an unlucky situation, part of football. Because of contact, he had a problem with his knee. The chances are he requires surgery so yes, it could be long term.
"It is a big blow for the players because they were really motivated, they were performing well and it is now time to focus on the guys who are fit because we have really important games now."
Both Danilo and Lawrence will be absent during an important period in Rangers' season, with Saturday's clash with Dundee followed by their final Europa League group-stage match against Real Betis next Thursday - a game in which the Gers can qualify for the knockout rounds with a victory in Spain.
Rangers will also have one eye on the Scottish League Cup semi-final against Aberdeen on December 17 as well as the next instalment of the Old Firm derby against bitter rivals Celtic on December 30.
Clement has also confirmed that Scott Wright, Ryan Jack and Nico Raskin will all miss out on Saturday due to injury, while Kemar Roofe is closing in on a return but is not ready to play 90 minutes.
Despite the absences of a number of first-team players, Clement believes his side are capable of success in both of their upcoming cup matches, but he has insisted that his focus on Saturday's clash with Dundee.
"It's a decisive game in the League Cup. The other competition will define where we go further, and that's already an important thing to know we're playing in Europe next year," said Clement.
"And of course we want to play in the Europa League, but I know a really good manager who put a tattoo on his arm when he won the Conference League so don't throw away that competition. We're going to go to Betis to try and finish first in the group.
"It's a marathon until the end of the season in the other competitions, and we also have the Scottish Cup in January. There are three competitions after this week which we'll still complete."
Rangers head into Saturday's contest with Dundee sitting second in the Scottish Premiership table, eight points behind Celtic who have played a game more.