The MLS club are said to have offered the Ivorian defender a deal until 2017.
Toure, 34, is out of contract at the end of the season but revealed last month that talks over an extension were ongoing.
Manager Brendan Rodgers is said to be keen to retain his services, in the knowledge that his developing squad would benefit from his experience, but the Ivorian himself remains hesitant over settling for a bit-part role.
“The club have to think and I have to think as well,” Toure told the Liverpool Echo in March. “It is a decision [whether to stay as a squad player or move for regular football] and that is why I need to think very hard and then take a decision and stick with it.
“It is very important to know what you are facing. Whether you are going to be playing or not. I need to do the right thing for me and my family.”
Now Toure is said to have options across the Atlantic, with a source claimed to be close to the Reds centre-back telling the Mirror: “New York Red Bulls have made an approach for Kolo. Once the time comes, we will decide what is best for him.”
With two Designated Players slots open following the offseason losses of the now-retired Thierry Henry and Tim Cahill, the Red Bulls have room to make a couple of big-name additions in the summer.
In addition, despite remaining the only undefeated team eight weeks into the 2015 MLS season, Jesse Marsch’s side are still in desperate need of some defensive reinforcements, although the head coach is keeping tight-lipped about their transfer plans at this point.
“We’re still evaluating,” Marsch told SBISoccer.com earlier this week. “I would say that, for me, the main focus is that we really like this group. If we add to it, we don’t want to add to it just because you have a DP spot or you have the ability to add.
“We want to make sure that everything we do here is very calculated, and regardless of it’s a DP or any player that comes in here, we want it to be the right fit.
“We want it to add to what we have because, like I said, we feel really good about the group we have and the way we’re playing and who we are. It’s a delicate balance, because you can sometimes subtract by adding, and we want to make sure that we add by adding.”
Toure, meanwhile, has featured 44 times for Liverpool since arriving from Manchester City on a free transfer in the summer of 2013, including 20 total appearances this season.
The former Arsenal man has performed well at times when deputising for the likes of Martin Skrtl but it remains to be seen whether he will opt to extend his Anfield spell or instead be tempted to take his wealth of experience elsewhere.