Walcott, who has also been linked with a return to former club Southampton, has slipped down the pecking order in Arsene Wenger's plans and has yet to start a Premier League game this season, appearing just five times.
Since Allardyce was appointed at Goodison Park, the Toffees have scored eight times in as many league games, prompting the former England boss to also snap up Turkey international Cenk Tosun last week in a £28 million deal.
"You're all aware of our interest in Theo," Allardyce told a news conference.
"If we're able to get that over the line, I'll be delighted.
"If we can get that all sorted out then that would be a great addition to 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.
"So, if that's at all possible, that would be very nice. It's not imminent, but negotiations are going on."
Walcott has 65 Premier League goals and 43 assists to his name in 269 appearances for the Gunners and Allardyce believes he could be the perfect tonic to his ailing front line.
Sam on Walcott: "It would be very nice if we could get it over the line but there is nothing imminent. It would be a permanent deal. We wait and see.”https://t.co/e0fD3TUrCd— Everton (@Everton) January 12, 2018
Arsene Wenger faced the media shortly after Allardyce and the Arsenal manager believes a transfer could help Walcott's chances of making England's World Cup squad.
Asked if the move was motivated by Walcott's international ambitions, Wenger said: "Yeah of course, everyone wants to play."
He added: "He's obviously very eager to make an impression. He's in the squad for Tottenham, we have to try and hope he can hit the ground running.
"We don't feed him too much information, hopefully his natural game is enough to make an impression and make us a goalscoring threat. It depends on the service he needs to be a goal threat."