The Saints have some cash to splash following the £75million sale of Virgil van Dijk to Liverpool.
And after ending a run of nine games without a win with their 1-0 FA Cup third round victory over Fulham on Saturday, Pellegrino is eager to strengthen his side.
Walcott, who has played just five Premier League games for Arsenal this season, prompting the 28-year-old to potentially reluctantly consider a move away from the club.
Pellegrino said: “I have been talking about him.
“He is a good player, but I don’t know with the negotiations, with the market, I imagine that this type of player would have a lot of possibility. For us it’s not easy.
“Hopefully we can bring somebody that can help us be stronger. This is our target now. We know the market is difficult, but we are working on that.”