So here is the breakdown based on my quick research:
2016-2017 Transfer Window
Son reportedly wanted to leave Spurs after his unsuccessful adjustment in the PL.
Liverpool wanted him since Son had already rejected Liverpool to join Spurs in 2015.
New Tottenham Hotspur signing Son Heung -Min reportedly rejected a move to Liverpool . According to Tom Collomosse of the Evening Standard , " Son’s camp have ...
bleacherreport.com
But Son wanted to go back to Bundesliga, and there was a rumour that he was wanted by Wolfburg.
2017-2018 Transfer Window
Son was 25 years old, and had just scored 21 goals in the 16-17 season. But he had 3 years left in his contract. The article below suggests that Man City was interested in him at this time, but his long contract combined with the Levy factor deterred them from seriously pursuing after him (City just signed Kyle Walker for a world-record 53 million pound deal for a defender during this window).
SAMI MOKBEL: Back in the Champions League with his first Golden Boot taking pride of place on his mantle piece, Heung-min Son couldn't be happier. Matters could have easily been different.
www.dailymail.co.uk
2018-2019 Transfer Window (2018 Asian Cup & Pending Military Conscription)
Son turned 26 in July of 2018. Many teams would be attracted to footballers at that age with only 2 years left in the contract, but big teams were not interested in Son due to his looming military service (about 2 years).
Had Son not won the Asian Cup to be exempted from the conscription later that summer, he
HAD to leave the Premier League before the 2018-2019 season to join a Korean domestic team to be eligible to play for the military football team to continue his football career. To be eligible for the military team, an applicant has to be 27 years or younger, and has played at least 1 continuous year in the domestic league immediately prior to joining the military team.
On July 23rd, Spurs gambled and gave Son a 5 year contract extension on July 23rd (which I believe is the main reason behind Son's unwavering loyalty towards Spurs). Son would not have left Spurs right after getting a 5-year contract extension and being excused from the pre-season to play for Korea in the Asian Cup (the club did not have to let him go since the tournament is not a FIFA sponsored event). Also, the Asian Cup ended on September 2nd, meaning the transfer window had been closed by then any way.
There was a rumour that Bayern was interested in Son for a January sweep after his military exemption. But Son gave an interview that he had recently extended his contract and was happy where he was.
Son Heung-min to Bayern Munich? | Spurs in for Arne Maier and Frenkie De Jong? | Nicola Barrela to Liverpool? | Aaron Ramsey to Milan?
www.theguardian.com
Reports from Italy recently emerged suggesting Bayern Munich are interested in signing Tottenham winger Heung-min Son.
bayernstrikes.com
2019-2020 Transfer Window
This is right after our incredible run to the Champions League final. So many teams would have wanted Son, but Son still had whopping 4 years left in his contract (and only 27 years at this time). So imagine the cost to buy him from Levy right after his CL semi-final performance. I saw a couple of news reporting rumours linking Son to Juventus, Liverpool, Napoli, but nothing concrete.
2020-2021 Transfer Market
A couple of teams were reportedly interested in 28-year old Son (just pasting some articles below, don't know how reliable these sources are but there are many articles out there).
Real Madrid, especially Zidane, were reportedly interested in Son, but their priority was Mbappe, and any deal for Son would have far exceeded 100 million pounds, which deterred any serious bid by them.
Tottenham star Son Heung-min has emerged as a Real Madrid transfer target.
www.express.co.uk
Bayern were interested in Son as well, but Son was too expensive for his age. Bayern had enough wingers and had a policy of not signing older than 25 years for a certain amount of fee.
The German and Bundesliga champions are targeting a move for Tottenham Hotspur star Son Heung-Min amid negotiations stalling over a new Spurs contract, according to reports.
www.dailymail.co.uk
I also read an article that the Club did not want to let go of the star players when we just missed out on qualifying for the Champions League, as it would give a wrong signal and lack of ambition to the supporters.
2021-2022 Transfer Market
Son renewed his contract on July 23rd, so no team would have come in to get him during this transfer window.