Tottenham and England legend Greaves, who sadly passed away last weekend at the age of 81 after a long illness, was the leading goalscorer in the English top flight in six different seasons during his career in the 1950s and 60s.
His record of 357 goals in 516 top-flight appearances is 74 more than Alan Shearer's current all-time Premier League goal record, and for almost half a century, it was the record for all of Europe's five major leagues until Cristiano Ronaldo finally surpassed him in 2017.
So, what better way to remember one of the most iconic goal machines of the English game, than by renaming the famous trophy in his honour
The Sun's Dave Kidd floated the idea in his column, and Hawksbee is all for it.
He said on Tuesday's Hawksbee and Jacobs show: "It's a great idea, in honour of Jimmy Greaves - is there a better time to do it?
"He maybe didn't get the honours he should have got when he was still with us, but posthumously it would be a really nice tribute.
"It's a brilliant idea and I hope the talkSPORT listeners get behind it, and the Premier League do as well.
"It's a pretty simple thing to do and I don't think you'll get any pushback from anybody.
"A lot of people in sad circumstances over the last 48 hours have seen goals and videos of Jim in action that they've never seen before, and they can see what an amazing player and amazing goalscorer he was.
"It's a great idea, so let's hope that it gets a bit of traction and can be made to happen!"
There have also been calls to honour Greaves with a statue outside the Tottenham Hotspur Stadium.
Spurs currently have a policy of no statues outside the ground and do not plan to change that, according to reports.
talkSPORT host Andy Goldstein backed the calls for a statue and told Spurs to rip up that policy.
He said: "If Spurs have got a policy on those statues, change the policy. Jimmy Greaves is Tottenham's greatest ever player, he's one of England's greatest ever players.
"Spurs at the moment, and have done for a long time, have been desperate for some decent PR.
"Tottenham's fans have suffered for a long, long time for various reasons. Not happy with the owners, Daniel Levy and certain managers, not happy with the time they've waited for a trophy, the list is endless.
"This is a great opportunity for Tottenham to have a tiny bit of good PR to get the fans saying 'you know what, that's a nice thing to do'.
"There are plenty of places to put this statue. Get the statue made."
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