Wolves are eyeing free agent Kei Kamara as they try to apply for a work permit for the striker.
Southampton striker Dani Osvaldo admits he would be keen to make his loan move to Inter Milan permanent.
And Wenger stressed the ongoing contract wrangle must be settled "one way or another" before the new year.
It was a rare start for the 23-year-old, who has seen his first-team chances restricted to mostly being utilised as an impact player from the bench.
The player's current deal expires next summer, with his representatives and the club still some distance apart over wages and Walcott also expressing concerns over opportunities in his preferred central role.
If no deal can be brokered before January, Walcott would be able to negotiate a potentially lucrative pre-contract agreement with a foreign club or just hold firm and walk away on a Bosman free.
While Wenger would prefer the situation not to get to that point, the Arsenal manager accepts a line must soon be drawn.
"My desire is to keep Theo. We do what's needed to keep him and hopefully [as] I told you many times, we can sort this situation out very soon - very soon is before the end of December, one way or another, that is for sure," said Wenger.
The Gunners boss was at Great Ormond Street Hospital on Thursday afternoon along with Jack Wilshere and Carl Jenkinson to help open the Arsenal Lung Function Unit.
On Walcott, he added: "I don't want to go into any details [over wage negotiations], but you can believe me [that] we do the maximum we can to keep our best players."
Like Walcott, Germany international Lukas Podolski, signed from Cologne in the summer, has expressed a desire to lead the Arsenal attack, rather than be deployed out wide.
Wenger, though, has no concerns over a player tactical revolt.
"I don't find many players who don't want to play central," he said.
"Podolski has 101 caps in Germany, and 101 on the left. I brought him as well with the idea that, when needed, he would play through the middle.
"I like the idea, but we will use him where he is useful for the team."
Wenger feels Arsenal have plenty of options in attack, despite having seen main striker Robin van Persie sold to Manchester United in the summer.
"It is not a problem - not at all. I believe it's important to have potential central strikers," the Arsenal manager added.
"We have Walcott, Podolski, of course Gervinho can play through the middle.
"We still have [Olivier] Giroud and [Marouane] Chamakh so it's good to have plenty of players who want to play there and [can] be capable to play there."
Goalkeeper Wojciech Szczesny is close to a comeback from his ankle problem and could be included in the squad for Saturday's visit of Fulham.
Midfielders Aaron Ramsey (groin) and Alex Oxlade-Chamberlain (hip) face fitness tests.
Wilshere, however, must serve a one-match suspension after his red card in last weekend's 2-1 defeat at Manchester United.
The 20-year-old has been called up to the England squad for the friendly in Sweden, after only recently returning from an injury absence of over a year.
Wenger has again urged caution on the midfielder's management.
"Jack is generous on the football pitch - he gives 100%. I feel physically, he is quite there - slowly he gets stronger in every game, that means his basic fitness is good," said Wenger.
"Sometimes when you accumulate games you need to have a little rest again, to build on that, to recover and then go even harder.
"For us, what is important to know is to stop him before he gets to that point."
Arsenal's charity partnership with Great Ormond Street Hospital in 2009/10 helped raise more than £800,000, with the new lung function unit part of that legacy.
The club have dedicated their Barclays Premier League fixture against West Brom at Emirates Stadium on December 8 to help raise money for more good causes as Wenger and his players donate a day's wages to The Arsenal Foundation.
Wenger said: "This facility at Great Ormond Street highlights the power that a football club can play in its local community and with our dedicated matchday on the horizon once again, I would encourage all supporters to continue showing their support as it really does make a difference."