Liverpool are also close to signing Crewe’s 16-year-old midfielder Dan Smith in a £400,000 deal, beating Manchester City to his signature.
For Kuyt, the decision to end his spell at Anfield a year before the end of his current deal would be an extremely difficult one.
He has made it known he still believes he has more to contribute at the highest level, and recently told Telegraph Sport that although he sees himself ending his career at Feyenoord — where he remains a hero – that was not in his immediate plans.
“I know I’m over 30 but I still feel physically very strong,” he said.
“I want to stay at the highest level for as long as possible, to get Liverpool back into the Champions League and to be part of the squad that plays in the competition.
"In the future, everyone knows Feyenoord is still a special place to me, but I am not thinking about anything but Liverpool now.”
Although the Rotterdam club have made no secret of their wish to take Kuyt back, it would require a significant cut in wages for that deal to happen this summer.
Feyenoord’s sporting director, Martin van Geel, recently suggested financial constraints meant an immediate move was “unrealistic”.
However, Liverpool’s willingness to listen to offers will certainly alert them and plenty of other clubs in Europe.
Fiorentina and Roma have also been monitoring Kuyt’s situation and his desire for European football would lead him to consider plentiful options.
For Liverpool to consider Kuyt’s departure comes as a surprise, given the lack of depth in their squad and greater congestion in their fixture list next season.
Firstly, with a year left, it is unlikely Liverpool would raise much in the way of a transfer fee given his age and length of contract.
On around £70,000 a week Kuyt is one of the squad’s top earners — he signed an extension last year – but expecting to replace a player of his pedigree at a low price would be highly ambitious.
Liverpool will also need a bigger squad due to their Europa League campaign, so offloading Kuyt would again seem risky given his experience.
The Dutch striker, signed for £11 million by Rafael Benítez in 2006, has been forced to readjust to being a squad player recently and was particularly disappointed to be left out of the Carling Cup final, when he came off the bench to score against Cardiff.
He is an extremely popular member of the squad due to his wholehearted style of play and knack of scoring in the biggest games.
His winner against Manchester United in the FA Cup was the most celebrated at Anfield this season and last year he struck a famous hat-trick against the champions.
He will be seeking clarification of where he will be fitting into Liverpool’s plans next season should he decide to see out the last 12 months of his deal.
Rodríguez’s exit will come as less of a surprise.
The Argentine has barely played all season, initially because he was overlooked for the new recruits and then because he has suffered numerous injuries.
Full-back Aurelio was surprisingly re-signed on a pay-as-you-play deal by Roy Hodgson in 2010 having been earlier released because of a chronic injury record. His bad luck with injuries has not been any better since.
Meanwhile, Daniel Agger and Glen Johnson are ready to give Liverpool a timely injury fillip before next week’s FA Cup semi-final with Everton at Wembley.
The duo hope to play in one of the next two Premier League games, either against Aston Villa on Saturday or at Blackburn Rovers next Tuesday.
The pair have been sorely missed in recent weeks as Liverpool have suffered six defeats in their last seven league games.
Agger, who was probably Liverpool’s player of the year before his recent absence, has been missing with a broken rib while Johnson suffered a calf injury after the Carling Cup final.