But the Hoops chief says the Republic of Ireland assistant manager faces competition from as many as nine other rivals to be named Neil Lennon's successor.
Ireland boss Martin O'Neill revealed on Thursday that the majority shareholder of the Scottish champions Dermot Desmond had held talks with his number two.
And now Lawwell has confirmed the 42-year-old former Manchester United skipper - who ended his playing career at Parkhead - is on a short list of candidates the Celtic board is now examining.
Asked how many names were in the frame, Lawwell said: "Between five and 10.
"We are looking at a number of exceptional candidates and we're going through a process. Hopefully it won't take too long.
"Clearly we want to do it as soon as we possibly can but these things you can't put a time on.
"We have got a number of excellent candidates, Roy is one of them and we'll see what happens in the next week or two.
"He's played here, he's a Celtic supporter, he's a first class guy and like the others are, as I said we have between five and 10 candidates.
"You just have to do your work properly and hopefully we'll get it done as soon as we can.
"As soon as we do that we will let our supporters know.
"You don't want to get it wrong by going too fast, you need to take your time and make sure you get the right guy."