Arsenal have already brought in France Under-21 attacker Yaya Sanogo on a free transfer and Arsène Wenger is armed with a £70 million budget to end the club's trophy drought.
Speaking in a question and answer session with ESPN FC, Wilshere said Rooney's winning mentality would be "amazing" for Arsenal.
"If it happens, it would be amazing," Wilshere said. "He (Rooney) is the type of player who can win you trophies - and just to see his name on the team sheet brings fear to the opposition. We could do with a little more of that."
The club have apparently had a bid rejected for Bayer Leverkusen's defensive midfielder Lars Bender today and Wilshere, 21, claims Arsenal are set to be more competitive in the transfer market.
He added: "You look at the type of players like Higuaín we are trying to bring in and you have got to be encouraged.
"I think we need a few more, not to step right into the team, but to add to the squad.
"We need our squad to be as deep as United's and (Manchester) City's. They can change their back four completely and still win. There is no way United would have won the Premier League trophy with 18 players."
Wilshere admitted: "It is no secret we need to win a trophy. How long has it been now, seven or eight years?
"I think once we win one, others will follow."