Arfield is valued at £3 million by the report, a good price for Rangers considering he was snapped up on a free transfer by Steven Gerrard last summer.
It is not thought Rangers are keen to sell the Canadian, not after his hat-trick for the club at the weekend.
There is an interesting backstory to the Arfield to Southampton rumours. One which dates back nine years.
Arfield was linked with a move to the Saints before, back in 2010.
Southampton were in League One at the time and Arfield was a young player at Falkirk.
Arfield rejected a proposed £500,000 switch. He did so because he did not feel he could live up to the fee.
He told the Daily Record in 2010: "To be honest, even with all the speculation, I didn't deserve a move.
"I didn't think the money getting mentioned was fair. In that period I wasn't worth that as a player. But I have a few months left this season and it's up to me to perform.
"It wasn't that I was affected by the speculation. But it's up to me to prove what kind of player clubs would be getting if they paid the money."
The decision worked out for Arfield. He eventually made it to England's Premier League, starring with Burnley.
As much as he is enjoying life at Rangers, if he does harbour any Southampton regret, he has a second chance.
Unless he decides to reject them for a second time.
Have something to tell us about this article?