The Toffees have started the season in poor shape, losing at home against Fulham and then earlier today being taken apart by Aston Villa.
Unai Emery’s men, who shipped five goals at Newcastle United last weekend, kept a clean sheet at Villa Park and ran out 4-0 winners over Everton.
Sean Dyche’s men looked all at sea and they are now intensifying their efforts in the transfer market.
Southampton recently rejected a £12m bid from Everton for Adams, but now the Toffees are back with an improved offer.
Everton have put £15m on the plate for the striker and will hope it can tempt Saints to sell.
Dyche will also hope the way his side have started the season will not put Adams off making the move to Goodison Park.
Everton are next in action against Wolves at home next weekend, before they then kick off their EFL Cup campaign away at Doncaster Rovers.