Southampton are said to be interested in both Mepham and Ollie Watkins, the Welshman's teammate who plays in attack.
But it is claimed their south coast rivals want a centre-back to complete their January business, which began with the £19 million capture of Dominic Solanke and continued with the striker's former Liverpool colleague, Nathaniel Clyne.
Bournemouth had a £10 million offer turned down for Mepham in the summer transfer window, according to the Sun.
Gift from the Gab?
It was reported last month that the new Southampton manager, Ralph Hasenhuttl, must sell before he can add to his squad in January.
And it remains to be seen whether the departure of Manolo Gabbiadini back to Sampdoria - at an alleged cost of around £11 million - will enable the Saints to counter Bournemouth's planned Mepham bid.
Bournemouth, Southampton or somewhere else? Where is Mepham heading this month?