Frenchman Robert is on the way out at St James' Park after a series of bust-ups with manager Graeme Souness, who he has publicly criticised.
He has spoken to Portsmouth and Toulouse already but Allardyce is on the verge of tempting him to the Reebok in a '1m deal with the offer of a three-year contract compensating for a pay cut.
The player's agent Willie McKay said: " I expect a deal to be completed on Monday."
Kyrgiakos has been a big hit at Rangers since joining them six months ago.
However, Rangers face a '1m bill from Panathinaikos to keep him and he could return to Greece unless Bolton decide to move in.
Middlesbrough and Charlton are interested in Bolton's transfer-seeking central defender Radhi Jaidi.
Wanderers insist he is not for sale but an offer approaching '3m could change their mind if they could bring in Kyrgiakos to replace him the squad.
Allardyce is more eager to bring players into the club than lose them at present after his disappointment at missing out on Liverpool's Dietmar Hamann.
That has been exacerbated by Vincent Candela, rejecting the chance to return to the Reebok in favour of a move to Udinese.
Young Bolton striker Ricardo Vaz Te was in the Portugual side who lost 4-1 to France in the final of the Toulon Under-20 tournament.
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If Allardyce can gain the respect and instill a season-long work-ethic in Diouf, then I'm sure he can work a similar miracle on Robert. There are no passengers at Bolton; and if Robert is transferred, he's going to have to be disciplined, get down to a season of hard work, and score some much needed goals. It would also be helpful if he kept his kit on and his mouth shut!
Laurent Robert has never really broke sweat at Newcastle, i would be shocked if he can be bothered to put in a shift at Bolton. Kyrgiakos looks like another Ibrox dud, in a very average Rangers team. But if anybody can get the best out of them Sams the man.