Best's contract is due to expire in June and Coleman has already resigned himself to the fact a new deal is not going to be agreed.
But his immediate future will be decided over the remaining 11 days and, as yet, nothing has been forthcoming.
"I have not had one offer, not one bid," the City boss told the Coventry Telegraph.
"They (Boro) are obviously interested in him so we have got to be pro-active and say, right, let's get our ducks in a row and say this could happen.
"Let's be honest here, if it doesn't happen now he is going to go in the summer because he is not going to sign a new contract with us. So let's make sure we get someone in before we get one out."
Coca Cola Ch'ship Promotion 09/10 Win Outright: Coventry 25/1