Browning and Galloway have both had fleeting chances to impress in Everton's first-team this season, but Bolton are confident in their chances of landing the duo on loan.
Neil Lennon wants to freshen up an ageing squad at the Macron Stadium and is said to feel that both Browning and Galloway would be good additions.
Everton could be keen on the move as it would give two of their up-and-coming prospects the opportunity to play regularly at a good standard.
Bolton will be one of the teams expected to chase a play-off place in the Championship next term, and that would give the Toffees youngsters a good platform on which to showcase their talents.
Browning and Galloway are considered to be two of the brightest prospects coming through at Goodison Park at a time when there is a clutch of exciting young talent at the club.
The likes of John Stones and Ross Barkley, who have both spent time in the Championship previously, appear to have very exciting futures at Everton, and Galloway and Browning will be hoping to follow in their footsteps.