The speculation has been such that it is almost taken for granted that Everton will sign at least one of them.
The two striking targets fro the Blues in this transfer window have become viewed as a binary choice: Olivier Giroud or Christian Benteke.
Such is the optimism which has been gained by the speed and decisiveness at which, pre-Gylfi Sigurdsson at least, Everton have signed players so far this summer, it seems a formality that the Blues could lure one or both to Goodison.
Giroud’s primary wish is to remain at Arsenal . He is settled in the capital and relished the fight for his place last season, eventually scoring 16 goals overall despite not starting regularly.
There are rumblings that he is not keen on moving North.
Of course his mindset might be altered by Alexis Sanchez’s future. Manchester City remain quietly confident they’ll get the Chilean before the window closes, Arsene Wenger is content that he’ll keep him.
If Sanchez stays it’s another obstacle to the first team for Giroud, a challenge intensified by the £46.5m arrival of Alexandre Lacazette.
The prospect of Benteke is more remote. His favourable statistics and ability to lead the line and capitalise on Gylfi-guided set pieces make him a great theoretical addition, but it could well remain an idea.
Once again there are suggestions he would rather not move back to the North-West after his frustrating spell at Anfileld.
It would be unlikely to be his decision anyway. Why would Crystal Palace countenance selling him? He is vital to their top flight consolidation and they could at the very least price him out of the market if they don’t simply hang up the phone.
One thing in Everton’s favour is the season beginning in a World Cup year. Giroud in particular will be better place to go to Russia if he is playing regular first team football; something Koeman would be more likely to promise.
But there are a lot of question marks. Enough to underline that a pursuit of neither is going to be simple.
Instead we should be prepared for a hitherto unknown name – perhaps a Steve Walsh special – to fill the vacancy.
Everton’s director of football will have a few lined-up that’s for sure. His challenge, though, will be to convince Koeman, a manager who is not averse to getting cold feet at Walsh’s penchant for taking qualified risks on damaged goods a la Ishak Belfodil. In the end Koeman simply decided that one wasn’t for him.
“I know which one I like but it’s difficult,” he said.
He wasn’t kidding.