The Hammers pair arrive at the Macron Stadium initially until January after travelling back from the Premier League club’s pre-season tour of Austria and Germany on Monday.
But it is understood there is an informal agreement between the two clubs to extend the loan in the New Year to the end of the season.
Tipped as future first team regulars at the London Stadium, Burke and Cullen have been on Wanderers’ radar all summer.
Phil Parkinson had hoped to bring them in at the start of pre-season but Hammers boss Slaven Bilic wanted to take a closer look at them on tour before granting permission for the move.
Burke is a centre-half who can also play right-back, and has been in the West Ham academy since the age of nine.
He played under Parkinson on loan at Bradford City two seasons ago and took the Bantams’ player of the year award but suffered an injury while on loan at Wigan Athletic last season, and was restricted to just 10 appearances.
In all he has played in 12 first team game for West Ham, making his Premier League debut in April 2015 against QPR.
Cullen also linked up with the Bolton boss at Bradford, and also spent last season on loan at Valley Parade in League One.
He has played four senior games for the Hammers but has also racked up nearly 60 appearances in League One and a call-up to the Republic of Ireland Under-21 team.
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