Walcott moved to the Emirates Stadium from Southampton in 2016 as a prospect for the future but has seen his playing time restricted in recent seasons.
The 28-year-old forward is indeed yet to start a Premier League game this season and it has been said that he would consider moving away this month in order to increase his chances of playing in the World Cup in Russia this summer.
Both Everton and Southampton have been linked with his services in recent days as a result despite Arsene Wenger having publicly stated that he would like to keep the Englishmen in his ranks beyond the end of the transfer window.
BBC recently reported that Everton could have the edge over Southampton in the race to sign the forward with the Toffees more likely to meet Walcott's £110,000-per-week salary and Arsenal demands - said to be around £20m ($27m).
Allardyce has now all but confirmed his interest in Walcott after admitting negotiations with Arsenal to sign the forward on permanent basis are ongoing.
"I think you are all aware of our interest in Theo. If that is possible to get over the line, I would be delighted but obviously there is negotiations that are happening at the moment," Allardyce confirmed in the press conference ahead of the Toffees trip to Tottenham Hotspur on Saturday [13 January].
"If that can all be sorted out, which is one of the most difficult things to do then that would be a great addition in my opinion to come and join us from an attacking point of view. Two to three reasons [for interest: goalscoring threat, pace, experience and he's good on assists from wide in terms of his crossing.If that is at all possible it would be very nice, but it is not imminent at the moment but some negotiations are going on. It is permanent we are hoping to negotiate. I don't think there is any chance of a loan."
Allardyce included Walcott in his plans during his short spell as England manager and would be delighted to coach him again at Goodison Park but the Everton boss didn't want to be too optimistic until a deal is completed.
"I don't get optimistic otherwise I get disappointed so we will wait and see. I will be very excited and enthused once somebody, if it is not Theo then somebody else, has signed on the dotted line," the boss added.
"In this window, or in any window, you don't get too optimistic because so many things can happen in a short space of time. You will never think a player is coming until somebody tells you he has actually signed on the dotted line."
Everton have already made one big money addition this month after paying around £27m to sign striker Cenk Tosun from Besiktas.
Allardyce is still keen on adding more new players to his squad ahead of the second half of the season but admitted that some members of his squad should leave before making more signing.
"It is essential that we trim the squad down if we are to bring more players in to adjust the balance of the squad. We would be hoping for some negotiations for some of our players if that became a possibility where the club were satisfied by what was offered from other clubs," the Everton boss added.
"We have had a number of enquiries for some players but none of those enquiries have hit anywhere near where our valuation might be."