The 44-year-old former Holland worldwide managed to guide the Royals to the Championship play-off final but they were beaten on penalties by Huddersfield Town.
The former Swansea boss also said that it was important to make clear that his departure from Leeds and signing with Middlesbrough were "two separate situations" and nothing in the process was premeditated.
Stam, 44, is on a shortlist of candidates drawn up by the club's owner Andrea Radrizzani whose arrival coincided with Garry Monk stepping down, according to the YEP.
But having received assurances from the player's uncle, club owner Steve Gibson that he does not have to sell his prize asset, Monk hopes to convince the centre-half to commit himself to next season's promotion campaign.
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"At Leeds, I was expecting to go forward but we couldn't find that". There's no getting away from that.
"If a big club in the Premier League knocks it might be a different story, the club is so realistic also, but as it stands now Jaap wants to finish his job at Reading", he said. That is the challenge that you want, I have signed for three years and I would like to make it at the first attempt but you know as well as me how hard it is to do that. "But one thing that this club is, is that it is equipped, it's ready and it's determined to bounce back at the first time of asking".
He replaces Marco Silva, who left to take over at Watford after Hull's relegation from the Premier League.
The club are believed to be edging closer to an agreement with Leeds United over left-back Charlie Taylor. "I've got an opportunity to show myself against and with the best players from our country".