Sheffield United will hold talks with James Beattie's agent regarding the possibility of a longer-term deal at Bramall Lane.
Former Stoke and Everton striker Beattie, 33, has yet to start for the Blades since returning for a second spell at the club on a short-term contract at the end of November due to fitness and injury, but Wilson is keen to tie-up a deal.
Wilson said: "There will be talks [on Friday] between the representatives. I've always been optimistic about it. He seems to have enjoyed himself here.
"Everybody has taken to him again on the second time round, so it's been very easy for him, his transition has been quite seamless to come into the club.
"I can't see any reason why he would want to go anywhere else, but I don't know what the options are. I'm not privy to what he and his agent talk about, so we'll wait and see. It possibly won't be resolved [on Friday], but he's got until the end of the month."
Beattie's game time at Bramall Lane has been limited, but his presence has had a positive influence on the club's other strikers and Wilson added: "His presence in and around the dressing room has been very helpful to us and his experience as we go along the season will be a big help as well."