The Scotland international scored both goals as the Saints managed to beat Leicester City 2-1 on Saturday to register their first win of the new Premier League season.
However, Adams’ Southampton future has been put under the scanner with Frank Lampard’s Everton showing interest in signing him before the end of the transfer window.
The Toffees have struggled for the want of goals this term, having managed to score just two in three matches so far, and they have put Adams on their list of targets.
They have already lost Richarlison to Tottenham Hotspur and want a striker to replace him, while Dominic Calvert-Lewin is injured.
Adams has been identified as a possibility but Southampton would only allow him to leave if they manage to get in a replacement they feel is of the same quality.
Benfica’s Goncalo Ramos is one of the players Southampton are targeting, but the Saints are not the only club interested in him.
Fellow Premier League clubs Newcastle United and Wolves are two clubs keen on Ramos as well.
Adams has also been linked with Leeds United, who want a striker, and big spending Nottingham Forest.