Cook was called up England boss Gareth Southgate for their upcoming friendly internationals, giving Burlingham another shot at cashing in with five years to spare.
The midfielder was included in Southgate's squad for their friendly against Brazil last November but did not play.
William Hill already seems resigned to making a payout.
"It looks like a case of when, not if his grandfather is repaid for the faith he showed in his grandson," William Hill spokesman Joe Crilly told British media.
"(Lewis) ... has been in fine form this season ...so let's hope he can ...bag a cap in the next few days."
England play the Netherlands on Friday evening and Italy on Tuesday.
Cook, who graduated from the Leeds United academy and captained England's under-20s to a World Cup win in June, has made 23 Premier League appearances for Bournemouth this season.
(Reporting by Shrivathsa Sridhar in Bengaluru; editing by John Stonestreet)
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