The Hoops received a major windfall this week after losing Kieran Tierney to Arsenal in a £25 million deal.
Lennon has only signed three senior players for Celtic so far this summer and, despite scoring 12 goals in their first two Premiership games, more reinforcements may be needed to stave off competition from Rangers for a ninth successive league title come May.
There are also some big-name players linked with a move away from Glasgow and, according to The Record, Lille are set to make an improved bid for Jozo Simunovic after a £4 million offer was recently knocked back.
And Sutton claims that the Northern Irishman should walk away from Celtic if £5 million is accepted for the big Croat.
He told The Record: "There’s no getting away from the fact Celtic need new faces though. If Lennon is not given serious money to do it then it will be nothing short of negligent and anyone on the board who thinks the squad has enough right now is delusional.
"You can’t have the manager working with one hand behind his back. There’s talk of interest in Jozo Simunovic but if Celtic accept £4m or £5m then Lennon would be as well walking away."
Simunovic has established himself as a very big player for the Bhoys since Lennon took the reins in February and losing him would undoubtedly be a blow.
Christopher Jullien, who fetched £7 million, has already joined, but Celtic have lost two centre-backs this summer - Dedryck Boyata and Filip Benkovic, who returned to parent club Leicester City.
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