Walcott is at the centre of a contract dispute with Arsenal, and with his contract expiring at the end of the season he is free to speak with foreign clubs from next week.
At the end of the game the 23-year-old England forward bowed and clapped all corners of the stadium while cries rang out of 'sign him up, sign him up, sign him up'.
"I think he loves the club and the club loves him," said Wenger. "My desire is to extend his contract - I think he belongs here and hopefully we can do it.
"We cannot ask much more than three goals and the performance he gave. He can play through the middle. I am convinced of that.
"He has qualities, and he can only improve the more he plays there. He has done very well and it just typifies what I think he can do.
"He can play in the other position on the left, right or through the middle and it is good because Giroud is basically a centre-forward only so I can play the two together."
New York Red Bulls striker Henry, who is training with Arsenal during the MLS off season, has been working closely with Walcott behind the scenes.
"I believe Theo takes advice from Thierry," said Wenger. "It's good when you need to talk to someone who has played in there (as a central striker)."
Walcott said he was delighted with his performance and that of his Arsenal colleagues.
"I've always said that I'm a striker," said the former Southampton winger who wears the same number 14 jersey Henry used to parade. "I'm not saying I want to play there all the time but it's an option for me and the team.
"It was a manic game today, it was so open and that played into our hands. Newcastle have got some top quality players and they play some good football at times but as soon as it went to five we could see some of their heads dropping.