The former Celtic mid-fielder hailed Lenny's job in guiding the Hoops into Thursday's draw in Nyon for the first time since Gordon Strachan led the Bhoys out of the group stages in 2007/08 as phenomenal.
But Grant believes any managerial move south of the border by the Northern Irishman to a club outside of the established top four, the two Manchester clubs Chelsea and Arsenal, would be a step down.
The 47-year-old is also concerned that Celts' success in Europe's top competition will mean that star guys like Victor Wanyama and Gary Hooper will be very much on the new year wanted list down south after their exploits in the Champions League.
Grant, who was back in Glasgow to promote ESPN's festive SPL coverage, said: "Neil's job in taking Celtic into the group stages has been phenomenal.
"At the start I don't think anyone gave them much hope and their progress has really surprised me.
"I thought maybe they'd get a result at Parkhead and maybe nick a draw on the road but that was the best I could see it being.
"Also, when they lost in cruel circumstances at Barca, I thought that may have knocked the stuffing out of them.
"But Neil has got his tactics spot-on and he has got his players performing with such belief. They have made it through to the last 16 draw and that will undoubtedly have put him on the radar of clubs down south.
"But what I would say to Neil is that outside of the top four in England, is there another club that is bigger than Celtic?
"The logic in leaving Celtic to take a chance on building things at another smaller Barclays Premier League club is not there.
"You look at the size of Celtic and their huge home gates, and there is no doubt that if they, or Rangers for that matter, went to England then in five or six years they would be challenging the Manchester clubs.
"Neil has a big decision to make, but is there any rush for him to go anywhere else when he is getting the results he is at Celtic?"
Grant added: "It is the same for the likes of Gary Hooper and Victor Wanyama. They have proven they can do it in the Champions League and it would not be a surprise if they attract interest in January from down south.
"Make no mistake, Celtic's achievements this season, including that incredible win over Barca, have meant they are being talked about in England in very positive terms."
Looking forward to Thursday's draw, Grant says there is no point in getting the crystal ball out.
He said: "For me, no matter who you get in the draw, they are going to give you problems.
"But the magic formula for Celtic is to make sure they are still in the tie after the first leg.
"After what Celtic did to Barca, there is not a team that will relish coming to Parkhead."