Millwall goalkeeper and Montenegro international Matija Sarkic, who was 26, has also died.
Southgate said: “I know that some of our players and staff actually played and worked with Matija as well so both pieces of news were received with shock and great sadness.
“Those boys that played with him are feeling that even more.”
Campbell’s son Tyrese – a striker who has spent the past seven seasons at Championship side Stoke City – called the pain of losing his dad as “indescribable”.
“He was the life and soul of every party and room he blessed, a one-in-a-million person that was loved by everyone,” the 24-year-old added.
Kevin Campbell made his senior debut for Arsenal in 1988 and scored nine goals in 22 appearances for the Gunners during their title-winning campaign in 1990-91.
He then helped the London club win the FA Cup and League Cup double in 1993, followed by the European Cup Winners’ Cup in 1994.
“Being a London boy and growing up watching him and Ian Wright playing together was brilliant for us youngsters,” former Crystal Palace striker Clinton Morrison, 45, told BBC Radio 5 Live.
“The way they scored goals and the way they celebrated, that’s how we were, looking up to those two players as idols.
“Kev was like that even when he finished football. He still had that bright smile and I only remember the good stuff, brightening up a room and being a top player and a great individual.”
Campbell moved into broadcasting after his playing career and worked for BBC Sport and Sky.
“He was adored by everybody,” said BBC Radio 5 Live presenter Mark Chapman.
“It doesn’t matter if you worked with him on 5 Live or supported Arsenal, West Brom or Everton, he was adored by everybody.
“He was always so positive and always had a smile on his face whenever he came into the 5 Live sport studio. And he was a heck of a striker.”
Campbell moved from Arsenal to Forest in 1995 and spent three seasons at the City Ground, scoring 22 goals in 39 games during his final campaign to help the club win promotion to the Premier League.
Following a short stint with Trabzonspor, he moved to Everton on loan in March 1999, scoring nine times in their final eight Premier League games to help the Toffees avoid relegation.
He was rewarded with a permanent deal that summer and stayed for more than five seasons.
Campbell’s playing career came to an end with spells at West Brom and Cardiff.
“I’ve been watching the TV and everybody who’s come on has said exactly the same thing, what a great guy he was,” said former Everton defender Phil Jagielka, who joined the Goodison Park club after Campbell had left.
“He made a lot of people laugh, he always had a good smile on his face. I’d seen him around and about the place at Goodison.
“I played with his son Tyrese at Stoke. It will be a really, really sad day for the family and all you can do is send your regards and, hopefully, they will get through this tough period.
“What a guy, what gentleman, what a character. I don’t think I ever saw him without a smile on his face and he was always one of those people to come up to you and give you a hug and have a little bit of a laugh and a joke.”
Former Tottenham keeper Paul Robinson, also speaking on BBC Radio 5 Live, added: “He graced our lives in the media and we all got to experience a little bit of his charm.
“We’ve lost one of life’s good guys today, a genuinely lovely man.”
Former England striker Alan Shearer told BBC One’s Match of the Day that he was “lucky enough to play” with Campbell for the England Under-21s “and work with him on bits in TV over the years”.
“He had an infectious personality and a great character,” added Shearer.
“You can see, with the outpouring of grief today, how much he will be missed. Sending our condolences to all his family and friends. Such sad news.”