The South London Press understands that the Terriers are set to return with an improved offer for the striker, who scored 23 League One goals last season.

May, who turns 31 next month, was signed from Cheltenham Town for a fee in the region of £250,000 in July.

The forward won EFL Player of the Year in February and also received a record-breaking volume of votes – 82 per cent – in taking the Addicks’ Player of the Year award.

May netted 27 goals in all competitions from 50 appearances and has notched 73 times in 204 League One matches in his career.

There is no certainty that the attacker will move on in the summer transfer window.

Not only would any interested club need to come up with an acceptable bid but Charlton boss Nathan Jones would also have to already have a suitable replacement in place.

May’s minutes were slightly reduced in the final fixtures – including playing just 45 minutes at Wycombe Wanderers on the final day – but he still finished two goals ahead of Colby Bishop to win the divisional top-scorer trophy for the first time in his career.

Jones won promotions at Luton Town from League Two and League One.

And he told our paper at the end of the season about building a squad that delivers success.

“We had a plan – exactly the same as here,” said the Welshman. “Tough decisions were made. We had to make difficult ones, evaluating people quickly.

“I had a conversation the other day about Luton fans who weren’t happy I released players when I did that first year – but if you asked them now if that was the right decision then I’m not even sure they’d blink in terms of that.

“We have to be clear on what we want to do. We have to be clear on certain athletes, first and foremost, what we want. If you’re not that athlete then we can’t take you forward, because this is what we’re going to do.

“Then it is the human being and the footballer – all those things have to come together.”