Bristol City and Brentford are also understood to have watched the player score a brace in Posh's 4-0 win over Wycombe Wanderers on Tuesday night.
Celtic have already added Polish striker Patryk Klimala to their ranks, giving them three first-team strikers in Odsonne Edouard, Leigh Griffiths and the new recruit.
Valued at £9 million, the 23-year-old was on Middlesbrough's radar late last year and if comments made by Peterborough chairman Darragh MacAnthony are anything to go by, the player is likely to stay with the League One side until the summer.
Peterborough director of football Barry Fry told Bristol Live: "Bristol City have shown interest in Ivan and we do have a good relationship with them, but there is zero chance of Ivan going anywhere in January.
"I have had eight Championship clubs ask about Ivan as well as a couple from the Premier League, but my answer is always the same. There is not enough money in this transfer window to even tempt us into talking about selling Ivan Toney."
Toney has had quite a nomadic career to date, starting off as a youth with Northampton Town before joining Newcastle United in 2015.
He managed just four appearances for the Magpies and spent time on loan at Barnsley, Shrewsbury Town, Scunthorpe United and Wigan Athletic, before being snapped up by Posh in 2018.
He hit 23 goals in 55 games during his first season at the London Road Stadium and has hit 19 already in all competitions this term.