Ipswich boss Mick McCarthy has offered a short-term contract to former West Ham midfielder Jack Collison.
Chairman Fawaz Al Hasawi has vowed to strengthen Nottingham Forest year on year, with loan arrivals possible.
Tonev hopes to follow the success of fellow Bulgarian Stiliyan Petrov and become a Parkhead legend.
Midfielder Petrov spent seven years with Celtic before taking the opposite path to Tonev, by moving south to join Villa.
As Tonev prepares for his big chance in the green and white hoops, the winger said: "Joining Celtic is unbelievable, and I feel really good about it.
“I'm delighted to have joined the club and I can't wait to meet up with my new team-mates and start playing for Celtic."
He added on Celtic's website: "I already knew all about Celtic. Everybody in Bulgaria knows Celtic.
"I spoke to Stiliyan Petrov yesterday and he was very happy for me. He wished me good luck and I hope I can do as well for Celtic as Stiliyan did."
Tonev has won 15 caps for Bulgaria and Villa hoped he could be a Premier League threat when Paul Lambert signed him from Lech Poznan in June 2013.
Petrov, who was on the Villa coaching staff at the time after an earlier battle with leukaemia, predicted his countryman could make a substantial impact, but the reality was that Tonev struggled to make an impression in the English top flight.
The move to Scotland could open the door to Champions League football, with Celtic having been handed a dramatic reprieve and a shot at Slovenian side Maribor in a play-off to reach the group stages.
Celtic manager Ronny Deila believes Tonev has more to offer than he showed in England.
Deila told Celtic's website: "Aleksandar's a very quick player and he has two good feet which he can shoot with.
"He has scored a lot of goals during his time at Lech Poznan and I think he can help bring the club forward with his qualities. We can also make him better."