The British practices are already down to the last four competitors and are battling against Morphosis with Gruzen Samton, and Rafael Viñoly Architects, to land the prize project.
The two phase scheme will add almost 130,000m 2of extra exhibition space to I M Pei's existing 1986 building (pictured).
There are also plans to create a new ballroom, banquet hall and 1,500-room hotel as part of the New York Mayor Michael Bloomberg's bid to bolster Manhattan's West Side redevelopment programme.
RRP has teamed up with local architects Fox & Fowle to draw up their entry and the winner is expected to be announced towards the end of the month.
Meanwhile, across the country in Iowa, Grimshaw has just been appointed to design a new ecological and sustainability centre similar to its Eden Project in Cornwall.
The Environmental Project in Coralville will house an indoor rain forest, integrated aquarium and exterior exhibits.
The AJ understands that a number of senior of architects who help develop the Eden Project will also work on this scheme.
Grimshaw director Andrew Whalley, said: '[It] is with this experience and resource that we will undertake The Environmental Project, and in doing so, create something that is truly unique, of international significance, and that casts a spell on all who visit.'