Ten Hag, 51, is among the runners and riders for the United vacancy, joining the likes of Mauricio Pochettino, Brendan Rodgers and Zinedine Zidane on the shortlist.
Solskjaer was sacked in the aftermath of a humiliating 4-1 defeat against Watford, ending his near three-year tenure at the club trophyless.
Dutchman Ten Hag is seen as one of the best up and coming managers in the game, having twice built Ajax sides that have thrilled the Champions League despite a lack of resources.
Sky Sports report that the Pep Guardiola prodigy is '100 per cent interested' in the job, but only in the summer, with Ajax adamant they wouldn't allow a manager to leave mid-season.
Following Sunday's 5-0 win over RKC Waalwijk on that keeps Ajax top of the Eredivisie on goal difference, Ten Hag was questioned about the possibility of moving to Old Trafford.
"I am focused on my team," he said. "We have a brilliant squad and we want to achieve a lot.
"We have a brilliant squad, we want to fight for titles. My focus is on Ajax, the rest only distracts.
"I expect from my players that they are focused on Ajax, so I have to give the right example for that too."
Clearly irked by the line of questioning, Ten Hag was insistent he wouldn't talk about the job, and claimed he hasn't heard from anyone in Manchester.
"You keep asking me this," he said
"It is a weird question. I heard nothing about it, so I can't think about it."
Ten Hag learned his trade alongside Guardiola whilst managing Bayern Munich's B team between 2013 and 2015.
He then returned to the Netherlands, excelling with Utrecht before taking over at Ajax, replacing Peter Bosz who had taken the team to a Europa League final defeat against Man United in 2017.
Ten Hag arguably one-upped his predecessor two seasons later when he took Ajax on an extraordinary run to the Champions League semi-finals, where they were knocked out by the narrowest of margins by Tottenham.
The Dutch giants lost the likes of Frenkie de Jong and Matthijs de Ligt following their heroic defeat, but Ten Hag has since rebuilt the team to the same standard.
Ajax were the first side to qualify for the Champions League knockout stages this season after picking up four wins from four, while Ten Hag has further impressed with league titles in 2019 and 2021.