Armstrong has scored 28 goals in his three-and-a-half seasons with Celtic, including 17 in 2016-17
Celtic midfielder Stuart Armstrong has been given permission to speak to Southampton after the clubs agreed terms for the player's transfer.
The Scotland international, 26, signed a new two-year deal last summer despite reported interest from Saints.
But Armstrong, who has been at Celtic since early 2015, is believed to be keen on the move to the Premier League.
Saints are ready to pay an initial fee of £7m which could rise further depending on various factors.
Southampton narrowly avoided relegation from England's top flight last season, finishing in 17th place.
Armstrong started his career with Inverness Caledonian Thistle and spent four-and-a-half years with Dundee United before joining Celtic.
He has won four Scottish Premiership titles, two Scottish Cups and two Scottish League Cups at Parkhead, playing a key role in the back-to-back domestic trebles under manager Brendan Rodgers.
Armstrong scored 17 goals in Rodgers' first season, leading to his first Scotland cap against Slovenia in March 2017.
Injury hampered his impact last season, though he played four more games for Scotland to take his tally to six caps.
Source: bbc.com