Crawford was forced off injured in Rovers' friendly with Sheffield United on Friday night.
There were initial fears the groin injury suffered was severe but boss Grant McCann says Crawford has a chance of being fit to face Southend United on Saturday.
"Ali is better," McCann said.
"We're hoping he will train on Tuesday and if he doesn't it'll be Thursday.
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"Thursday would be a cut off for me.
"We want him to get two or three days of training in before we go to Southend.
"Hopefully Joe [Sharp, physio] can work his magic and get him out on the grass.
"Ali said it when he came off, and it was more of a precaution with us not wanting to take risks in pre-season.
"He over-stretched when he took a shot in a great little move."
Alex Kiwomya has overcome an injury to a gluteal muscle and will be fit for the trip to Roots Hall.
"Alex trained on Monday and looked great," McCann said.
"He;s had a great pre-season and I've been really impressed with him.
"He's missed maybe two games, with the birth of his little girl as well.
"I've been really impressed with Alex. He's the sort of player who has got so much pace he's always likely to feel something somewhere and pull a hamstring.
"We've just got to make sure we be careful with him."
Rovers' main doubt for the weekend is right back Mitchell Lund who is suffering with an ankle problem.
"The only one we've probably got a real doubt on is Mitchell Lund," McCann said.
"We'll just see how he is for the rest of the week.
"It's his ankle. He's been struggling a little bit with it.
"We may need to refer him to a specialist.
"It's not going to be an operation, it's just to get a second opinion on him.
"We may need to inject him just to make him better.
"It's nothing serious. He isn't going to be long."
Issam Ben Khemis has been ruled out due to a suspected concussion with Rovers sending the midfielder for further tests.
McCann said: "Issam Ben Khemis is another one. He didn't play on Monday night.
"He took a whack from Andy Butler in training a couple of weeks ago before the Birmingham friendly.
"He played in that game but came off ill and he's been getting constant headaches ever since.
"Joe sent him for a CAT scan today because we can't take any risks."