Arsene Wenger's contract with the Gunners expires next summer and the 66-year-old is yet to make a decision on his future.
The Sun says the Arsenal hierarchy admire Howe's diligent approach to management and believe he can be moulded into a direct replacement for Wenger should the Frenchman decide to step down.
Wenger said last month that his future at the club will be decided by this season's results.
He was criticised by fans for failing to adequately reinforce the squad after Arsenal lost their opening match of the season against Liverpool at the Emirates Stadium on 14 August.
The north London club have since signed Germany defender Shkodran Mustafi and Spanish forward Lucas Perez for a combined £52m ($70m).
The Gunners have not won the Premier League title since 2004.
"My contract runs until the end of the season. What I will do after will depend a little bit on how the season goes," Wenger told the BBC on 10 August.
The Frenchman was linked with the England job before Sam Allardyce was confirmed as Roy Hodgson's successor.
He added that there was no chance of him taking over the Three Lions while he was still under contract at Arsenal.
"I was not available and Sam Allardyce has signed for two years, so the question is not of any actuality at the moment," the 66-year-old said.
Wenger agreed to loan Jack Wilshere to Bournemouth for the season and is understood to be an admirer of Howe.
"Jack left because he wanted to get more first-team football to get his England spot back," the source said.
"He loves Arsenal and wants to remain and the club and the chats he has had with the club have been very positive on both sides.
"Jack is desperate to stay and prove himself to everyone what a top-class player he is. The experience about Bournemouth should serve him well."