The 27-year-old came off the bench to score his side's winner against Sheffield Wednesday in the Championship on Friday night.
Adams has now managed 32 goals in 146 matches for Southampton in all competitions, including 25 Premier League goals.
According to The Sun, Palace boss Roy Hodgson has identified the Scotland international as the perfect replacement for Jean-Philippe Mateta, who is preparing to leave Selhurst Park.
The report claims that Wolverhampton Wanderers and Bournemouth are also keeping a close eye on Adams, with the forward being allowed to leave for the right price this summer.
The forward is believed to be valued by Southampton in the region of £15m.
Meanwhile, according to Sky Sports News, Burnley boss Vincent Kompany is keen to bring Manchester United defender Alvaro Fernandez to Turf Moor this summer.
The 20-year-old is back at Old Trafford following a successful loan spell at Preston North End during the 2022-23 campaign, where he made 42 appearances in all competitions.
Fernandez has been involved in Man United's preparations for the new campaign, but his route into the first team is blocked by Luke Shaw and Tyrell Malacia.
Kompany is said to be a huge admirer of the Spaniard and is keen to open discussions over a permanent move to the Premier League side before the end of the transfer window.
Fernandez represented Real Madrid before his switch to Man United in 2020, but he has not yet made his first-team debut for the 20-time English champions.
The defender was not in the squad for his side's pre-season clash with Lens on Saturday afternoon.
Staying with Man United, Hull City are allegedly showing an interest in signing Red Devils goalkeeper Matej Kovar before the end of the transfer window.
The 23-year-old has been involved for Man United during pre-season, and he was on the bench for their contest with Lens at Old Trafford last time out.
There is seemingly an opening in the goalkeeping department at the English club this season, with recent reports claiming that Kovar could be the number three behind Andre Onana and Tom Heaton.
However, following a successful loan spell at Sparta Prague during the 2022-23 campaign, Kovar is believed to be keen to depart on a permanent basis this summer in search of regular action.
According to The Sun, Man United are open to a sale, and Hull are firmly in the picture for his signature.
The report claims that the Tigers would initially be looking to sign the former Czech Republic Under-21s international on loan, while Sparta and Red Star Belgrade are also interested.
Kovar made the move to Old Trafford in 2018, but he is yet to make a first-team appearance for the club, spending time out on loan with Swindon Town, Burton Albion and Sparta.