The two clubs are negotiating a deal for the Newport-born shot-stopper, who turns 21 next month, for a permanent move away from the Montgomery Waters Meadow.
Former Premier League side Hull were last week linked with a move for the Town academy graduate and, according to sources in Hull, the Tigers are now ironing out the finer agreements of the transfer.
Burton still has a year remaining on his deal at Shrewsbury.
But the youngsters has found himself down boss Paul Hurst’s pecking order, with both Manchester United loanee Dean Henderson and former Walsall man Craig MacGillivray arriving this summer.
The future of fellow shot-stopper Jayson Leutwiler, Town’s No.1 for the three previous seasons, is also unclear.
Hurst has admitted it will be difficult for the Canadian, who has been away playing in the Gold Cup, to force his way into the first-team plans.