Root is just 2,457 behind Cook's total of 12,472 and is only the second player after Cook to score 10,000 runs in the format for England.
"He is a pleasure to watch, the most complete England batsman I have seen," Cook told BBC Sport.
"The person who could play the most incredible innings was Kevin Pietersen, but for the most complete batsmen in all three forms, it's Root. His consistency is incredible.
"He is so hard to tie down. I had to grind my way to 30, it always took me what felt like two hours. Because Joe has got so many low-risk scoring options, pretty much through 360 degrees, he will often get to 30 off 40 balls."
If Root continues in the form he displayed at the opening Test against New Zealand last week then he will certainly have a chance of not only overtaking Cook but challenging the great Sachin Tendulkar for most runs in history.
With that being said, we take a look at the top 10 players with the most runs in Test cricket.
Sachin Tendulkar - 15,921
The Indian legend made his debut at the age of 16 in 1989 and continued until 2013. During his career, he scored 51 centuries and 58 half-centuries.
Ricky Ponting - 13,378
The Aussie is in second spot. Ponting was an international Test player for 17 years from 1995 to 2012 and is the highest run-scorer across all three formats (ODI and T20) in Australia's history.
Jacques Kallis - 13,289
The South African is an all-time great. Not just based on his runs scored, but also the ability to take wickets too with 289.
Kallis got to three figures on 45 occasions with an additional 58 half-centuries.
Rahul Dravid - 13,288
India's Dravid made his debut in 1996 against England and ended his career with a fine average of 52.31.
Alastair Cook - 12,472
Cook is the only Englishman in the top 10 and considering he opened the batting, it is an incredible achievement.
The Essex star - who retired in 2018 - is the only player to score a double hundred in his first and final Test at international level.
Kumar Sangakkara - 12,400
The former wicket-keeper batter managed 134 Tests for Sri Lanka and ended up with an awesome average of 57.4.
Brian Lara - 11,953
In 16 years, Lara finished with an average of over 52 but it was the style in how he played the game which was more impressive.
England fans will remember him for his time at Warwickshire - where he scored 500 runs in one innings - as well as becoming the first batter to hit 400 runs in a Test match in 2004.
Shivnarine Chanderpaul - 11,867
In eighth spot is another former West Indies star. Chanderpaul was a player you could rely on to stay at the crease, steady the ship and score runs at the same time.
He scored 30 centuries, 66 half-centures and had an average of 51.37.
Mahela Jayawardene - 11,174
The second Sir Lankan on the list - who also made the top 10 for the most runs scored in ODIs.
Jayawardene's top score in Test cricket was 374 against South Africa in 2006.
Allan Border - 11,174
The legend finishes the list inside the top 10. Border - who played 156 Tests - held the record for most runs until Brian Lara overtook him in 2006.
He was the first player to score 11,000 runs, play 150 Test matches and in 2000, had a medal created in honour of his name.
The medal is for Australia's most outstanding male cricketer of the year. Mitchell Starc was awarded it this year while Ashleigh Gardner was given the Belinda Clark Award.