A number of Premier League clubs, including Norwich City, West Ham and Everton have credited with interest in the 24-year-old.
But according to Football Insider, the Saints feel they are in pole position to land Armstrong following breakthrough talks over a £10million deal.
The Magpies, who sold their academy graduate to Blackburn Rovers in 2018 for £1.5m, installed a sell-on clause in his contract.
Armstrong scored 28 goals in the Championship last term – bettered only by Brentford’s Ivan Toney – as he enters the final year of his deal at Ewood Park.
‘That would be a great bit of business’
Former England goalkeeper Paul Robinson says tieing Jacob Murphy down to a new long-term contract “would be a great bit of business”.
Murphy, 26, revived his Newcastle career last season after excelling in a right-wing back for Steve Bruce’s side.
He initially struggled following his £12m move from Norwich in 2017 but is now set to be handed a new deal on Tyneside as a reward for his much-improved form.
“That would be a great bit of business if they can get him tied down to a new deal,” Robinson told Football Insider. "He did well last season when he came into the side.
“It doesn’t sound like Newcastle have much money to spend this summer so players like that you need to keep.
“He is a proven Premier League player and he’s versatile, which is always a plus.”