The duo have worked with Phil Parkinson before at Bradford City and are said on the Whites’ shortlist if the Hammers decide to send them out for more experience.
Parkinson has marked out Cullen as a main target if the Hammers decides a year in the Championship would be better for his development.
Defender Reece Oxford has already been loaned out to German giants Borussia Monchendgladbach and they will loan out other promising youngsters who may not get a chance to play first team football.
Bolton would be able to offer regular football and have the fact Parkinson is well-known to both players counting in their favour. Burke played under the Bolton boss at Bradford two years ago and won the Bantams’ player of the year award aged just 19.