The Scottish champions can expect to receive an initial fee of around £7 million for the Scotland international but this could be augmented with add-ons.
It is understood his former club, Dundee United, will also receive a percentage of the transfer fee.
Armstrong, 26, signed a new two-year contract with Celtic last summer despite strong speculation linking him with a move to the St Mary’s club. But the player has long held an ambition to play in the English top flight and he now appears to poised to fulfil them. Southampton narrowly avoided relegation last season under the stewardship of Mark Hughes and are looking to strengthen their squad for the news season.
Inverness-born Armstrong joined Celtic in February 2015 alongside his Dundee United team-mate Gary Mackay-Steven after making 150 appearances for the Tannadice club and helping them reach two domestic cup finals.
He helped Ronny Deila’s Celtic win the league title that season but was cup-tied as his new club defeated his old one in the League Cup final.
The Parkhead side retained the championship the following year under Deila but it was under Brendan Rodgers in season 2016-17 that Armstrong really flourished.
A driving presence in the heart of the midfield, he scored 15 league goals in 31 games as Rodgers’ side won the domestic treble without losing a match. His form earned him a first Scotland cap, against Slovenia.
Injuries hampered his progress last season but he was still an important player as Celtic clinched a second successive treble.