Drinkwater was an unused substitute against his former club Leicester last Saturday and missed the midweek Champions League win over Qarabag of Azerbaijan.
Chelsea manager Antonio Conte said: "He had a muscular problem in his calf
I think it's very difficult to see him before the international break
He needs time to recover.
"It's a pity because he was starting to work and to improve his physical condition.
"Now we have to move all these situations
And in two, three weeks, four weeks, I don't know."
Drinkwater joined on transfer deadline day for a reported fee of Â£35million to be reunited with former Leicester team-mate N'Golo Kante
The midfield pair were key in the Foxes' 2015-16 surprise title win.
Chelsea play Arsenal on Sunday in the third of seven matches in September.
The Blues will have had more than 48 hours additional rest over the Gunners, who beat Cologne in the Europa League on Thursday night
Conte played down the extra time between games.
He added: "I watched the game and (Arsenal boss Arsene) Wenger rotated a lot of players, a lot of players rested for the game against us
I think it's not an advantage."
Chelsea's 3-0 loss at Arsenal in September 2016 prompted a change to a 3-4-3 formation and a 13-game winning run which led to the Premier League title triumph in Conte's first season.
Arsenal have not won at Chelsea since October 2011 and have won two of 21 games against the established top six since the start of the 2013-14 season.
But the Gunners denied Conte the double by winning May's FA Cup final and then won the Community Shield at the Premier League champions' expense, albeit on a shoot-out.
Conte was at pains to point out both Arsenal's wins came with Chelsea reduced to 10 men - Victor Moses was sent off in the FA Cup final, Pedro in August's season curtain-raiser - decisions which still leave the Italian feeling aggrieved.
"We finished two games with 10 men," Conte added.
"In both situations I think you have to see very well if there was the red card or not.
"I hope to finish the game with 11 players
Then during the game I think the best will win."
Eden Hazard could start for Chelsea for the first time this season after two substitute appearances
The Belgium playmaker suffered a broken ankle in June
Conte said: "To have this type of patience for him is paying off
He's very close to be totally fit."
Conte is likely to make a number of changes after rotating his players for the Qarabag contest, with four games still to play in September.
Heightened security is likely at Stamford Bridge in the wake of the Parsons Green explosion, which is being treated as a terror incident.
Conte opened his press conference by saying his thoughts are with those affected.
Source : PA