The White Rhino, 33, has been contacted about a possible bout with the New Zealander, also 33, after Allen beat Johnny Fisher, 26, in their rematch last month.
Speaking on his YouTube channel, Allen said: “We have been approached for a fight to see if I was interested in boxing Joseph Parker.
“Obviously I said yes, absolutely, I think he’s a fantastic fighter, one of my favorites, a personal favorite of mine. I think he’s brilliant.”
Allen recalled watching Parker beat Andy Ruiz Jr., 35, in their first fight for the WBO heavyweight title in 2016, and said it would be “life-changing and life-affirming” to get in the ring with Lupesoliai La’auliolemalietoa.
He said: “I remember watching him fight Andy Ruiz for the world title.
“They’ve asked if I have any interest so I said to [promoter Eddie Hearn] as much as I would like to win the British title a fight with someone of Joseph Parker’s ability, name, and stature, for a proper title would be life-changing and life-affirming I think as well.
“I could die a happy man if I boxed for a big title against someone like that.”
While Allen is raring to fight Parker, the New Zealander has recently been given the green light to get in the ring with Derek Chisora for the third time following his bout with the winner of Oleksandr Usyk vs Daniel Dubois - which is set for September.
When asked whether a third Chisora fight was on the cards, WBO president Gustavo Olivieri told Sky Sports: “Yes, Parker may defend his interim title against any of the top-15 WBO world-rated contenders.”
Parker’s promoter David Higgins revealed the boxer was open to setting up a trilogy fight with Chisora.
He said: “I'm sure he would be [open to a third fight], yes.”