The 22-year-old is closing in on fresh terms, the Mirror reports, having been Manchester United's leading target in the position over the summer, with Michael Carrick keen to add competition for Luke Shaw. United ended the window without strengthening there after Newcastle United made clear they would not sell at any price.
The Magpies took a firm stance on Hall having already sold Sandro Tonali, Anthony Gordon and Bruno Guimarães in recent windows, and they have since gone further by moving to tie him down with a new deal running until June 2031.
"Without speaking about his contract, I can just say that I'm in love with what I've seen so far from him," Jaissle said, speaking ahead of Newcastle's Premier League match against Leeds United on Monday.
"It's an amazing player with huge potential still, and this makes me so happy to, hopefully, have him here for the next years in the long term. He's fitting so good to this identity, how we want to play football, and again, he's doing an excellent job."
Hall, who joined Newcastle from Chelsea for a reported £28m two years ago, has played every minute of the club's games so far this season. He has provided assists in each of the last two matches, a 2-2 draw with Bournemouth in the league and a 1-0 win over Millwall in the Carabao Cup.
Newcastle have also been boosted by Tino Livramento's return from injury against Millwall.
"I'm happy for him that he got now 45 minutes on a high level against a strong opponent in difficult circumstances playing there in Millwall," Jaissle said.
"That makes us proud to see him already on a high level, it's never easy coming back from a long-term injury then performing already mature whilst physically already on a good level. I'm also satisfied and happy that he was dealing with the 45 minutes really good, he responded well. It means also our performance department did an excellent job."
United, meanwhile, will have to look elsewhere for left-back reinforcements in the January transfer window. With Shaw the only senior specialist in the position and Harry Amass not yet deemed ready, Carrick will continue to rely on Patrick Dorgu, Diogo Dalot or Noussair Mazraoui as cover — a need underlined after Shaw missed Wednesday's training session to manage his workload ahead of Thursday's Champions League tie against Sabah and Sunday's derby against Manchester City.