Allardyce said on BBC Five Live's Sportsweek programme: "We would like Roy very much. I hope he may glance at us or happen to see over the next couple of days how we played against Arsenal.
"He will have seen a winning team and that may persuade Roy to join us. But we can only decide that if his representatives give us a ring. We can't do any more than we did against Arsenal and if Roy was out there I'd ask him to have a look at that game and ask himself whether Bolton was a good club to join.
"I think he would really enjoy a stint with us. I have only registered an interest with his representatives, but many, many clubs have done that."
Allardyce added: "I am sure his people will ring us one way or the other and either say they'd like to go further or he has decided to go somewhere else.
"We have an advantage because he lives in the north west - unless Roy wants to up sticks with his family and move away from the area, we are the perfect club for him.
"There are many other clubs in many countries who would take him and are bigger than Bolton if he does want to move from the area. But if he wants to stay in the north west then we are a perfect choice."
Madrid are reported to have offered Keane an 18-month contract, and Allardyce said: "Keane for Real Madrid, yes that would be interesting. He is such a good player and although people have gone on about his age and injuries, that is not a concern for me.
"It has been broken bones that have been the problems for the most recent injuries. Those have been the only injuries he has had in the past 12 months.
"There is life in the old dog yet, certainly the experience he could bring and the fact that he is a winner would be great for a club like us."
Meanwhile, Charlton, Manchester City and Bolton are reportedly considering a move for free agent Giovane Elber.
The Brazilian was released by Borussia Monchengladbach on Friday by mutual consent and is free to sign for another club.