The Hoops are understood to have offered the 15-year-old the chance to come and train at Lennoxtown ahead of making an offer.
Spurs were believed to have stolen a march on their rivals by offering Allen a trial - Liverpool, Manchester United, Manchester City, Leeds and Leicester were all monitoring the teenager's progress.
The champions added two youngsters to their development squad last month, bringing in goalkeeper Tobi Oluwayemi and Norwegian defender Leo Fuhr Hjelde on three-year deals.
Allen has already made his debut for Linfield, and is a Northern Ireland Under-17 international able to operate at centre-forward or as an attacking midfielder.
He would almost certainly play in Celtic's reserve side to begin with but given Neil Lennon's tendency to reward highly-rated youngsters could see Allen handed a spot in the first-team squad in the not-too-distant future.