Walcott's career has notably stalled this season after 12 years at Arsenal, with the 28-year-old having yet to start a Premier League fixture in 2017-18 and generally now restricted to outings in the Europa League and domestic cup competitions.
Indeed, he has made just six substitute cameos in the top-flight since August and went a month without making a matchday squad between early November and December despite recovering from a virus.
Such a reduction in first-team opportunities in a World Cup year has led to speculation that Walcott could seek pastures new his month and the BBC claimed last week that both Everton and former employers Southampton had engaged in high-level talks regarding a transfer.
Arsenal are said to want a permanent £20m ($27.5m) deal for the 47-cap England international - now into the final 18 months of his contract - having rebuffed several loan enquiries, with the Toffees believed to be better-equipped to match his £110,000-per-week salary.
Allardyce later confirmed those negotiations as he hoped to make a "great addition" and continue to strengthen an attack left bare following the summer departure of Romelu Lukaku to Manchester United having already spent £27m to acquire Turkish striker Cenk Tosun from Besiktas.
'Big Sam', who fielded the player as a second-half replacement in his sole game in charge of England against Slovakia in 2016, pointed to Walcott's familiarity with the former Three Lions coaching staff - including Sammy Lee, Craig Shakespeare and Martyn Margetson - as an obvious plus.
Walcott has typically fancied himself as a central striker throughout his career. However, Allardyce does not appear to be guaranteeing him that sort of role in the event that Everton do manage to see off competition from the struggling Saints, who nurtured the player through their academy before losing him as a 16-year-old for an initial £5m fee in 2006.
'I don't know where I'll play him until we get that deal done," Allardyce was quoted as saying by The Mail. "When that deal's done I can have that conversation with him. I see him playing anywhere across the front three and I'm presuming if he's not comfortable with that then he'll tell us and it won't happen.
"We'd have that conversation if we were going to secure that transfer and we'll see where it goes from there."
Addressing Walcott's future last week, Arsenal boss Arsene Wenger reiterated that he wanted him to stay but admitted that he could not offer any assurances over playing time.
Allardyce is also believed to be looking to invest in a new left-back over the coming fortnight and has said that it is "essential" that out-of-form Everton, winless in six after suffering a fourth straight defeat against Tottenham Hotspur on Saturday (13 January), trim the current squad, though so far approaches for many current players are said to have fallen well short of their valuations.
"Any manager would say you'd always need more and it depends where we want to go in this window," he added.
Ross Barkley signed for Chelsea in a £15m deal on 5 January, while Kevin Mirallas joined Gethin Jones, Liam Walsh and Harry Charsley in leaving on loan last week. Antonee Robinson also extended his stint at Bolton Wanderers until the end of the season.