The spat began when Lukaku said in an interview in December that he had not spoken to Mourinho in two months and was not ruling out a permanent move to Goodison Park.
Mourinho, unable to resist, responded with a barb of his own, saying:
Romelu likes to speak. He's a young boy who likes to speak. But the only thing he didn't say is why he went to Everton on loan.
That's the only thing he never says. And my last contact with him was to tell him exactly that. Why do you never say why you are not here? So it's something he should (talk about) because he likes to speak.
It's something he should say, the reason why he's not with Chelsea and is in Everton.
Lukaku sought to calm the tensions by later insisting that he had no problem with Mourinho, though he also insisted that he DID explain to the Portuguese why he wanted to go out on loan.
He wanted me to stay and asked me why I wanted to leave.
In some interviews my words were twisted, but the truth is that he accepted to let me leave.
He did understand what I wanted. That's what I really appreciate in him.
The mysterious relationship between the two has been magnified by the fact Chelsea's remaining strikers - Fernando Torres, Samuel Eto'o and Demba Ba - have scored just 11 league goals between them this season while Lukaku has scored nine on his own.
Those goals have helped lift Everton into contention for a Champions League place and their remains a hope that should the Toffees finish in the top four, then a permanent deal for Lukaku may well go through.
A lot of things will need to fall in place for that to happen, not least the small matter of £20million that Chelsea will ask for (is THAT what the Jelavic and Fellaini money is being saved for? I won't go into that now...).
But one of them, a willingness by Chelsea to sell, may come to pass.
Speaking ahead of Monday's huge clash at Manchester City and discussing his summer transfer plans, Mourinho hinted that Lukaku may well be sold:
One to leave, two to leave because they want to play more because one of the kids feels it’s better to go and play on loan somewhere.
As I mentioned, Chelsea's willingness to do a deal is only the first hurdle, and even if Lukaku is keen on a move, the elephant in the room is Everton's lack of finance.
If indeed there was cash spare from the Fellaini deal and Roberto Martinez is free to spend all of the cash received for Jelavic, then - in theory at least - enough cash should be there for a sizeable down-payment.
We know though that a black hole exists somewhere at Goodison Park that swallows up transfer fees almost as soon as they are received, meaning players sold does not automatically equal cash to spend.
If we did manage to pull off a deal it would almost certainly mean Lukaku would be the only permanent signing of next summer (bolstered by loan signings of course). Is that a risk worth taking? Or should Everton spread the risk by signing several cheaper players? Have your say below.