The Magpies signed Woodgate for a reported £9million when they were competing at the top end of the Premier League table under Sir Bobby Robson.
The then 23-year-old centre-back had already been capped by England and established himself as one of the best young defenders in the country while at Elland Road.
But with Leeds struggling both on and off the field, Woodgate reluctantly made the switch to Tyneside.
"I didn’t want to go to Newcastle,” he admitted. “But Bobby Robson was manager, Alan Shearer was a legend, Gary Speed as well.
"They had a young team so I thought let’s do it even though I didn’t want to leave Leeds.”
Speaking on the Original Penguin X Campaign Against Living Miserably Under The Surface podcast, Woodgate added: “It was tough as you want to be successful in your career and want to play with the best players.
"But they all started leaving [Leeds]. Robbie Fowler went, Robbie Keane went and I knew they were going to get relegated, and they’re only back up [to the Premier League] two years ago.
“We saw the downfall from being in a Champions League semi-final – I’m over the moon they didn’t get relegated this season.”
Woodgate is often regarded as one of Newcastle’s best defenders in the Premier League era, but his time at the club – much like his overall career – was blighted by injury as he was limited to just 37 appearances in all competitions over two seasons before making a £13.4million move to Real Madrid.
“When you have injuries, there’s nothing worse as a footballer as you just want to play,” he continued.
"It seemed to be a snowball effect for me, as I’d get one injury and that would lead to another one, but it all came down to my back as I had an unstable segment in my back.
"Even though I didn’t get a massive amount of pain in my back, it was making me pull my hamstrings, my quads, groin injuries. But it was tough to take, especially as you know you could give a lot more, and had a more successful career without the injuries.”