The marquee arrival of the Valencia striker for a reported fee £27m plus add-ons would significantly eclipse the previous record of £18m paid out to bring Rio Ferdinand to the club in November 2000.
Rodrigo, 29, who spent a brief loan spell in England at Bolton Wanderers in 2010-11, underwent a medical in Spain on Wednesday morning after a deal was agreed in principle the previous evening.
Following completion of all the relevant documentation and barring any late hitches, Rodrigo is likely to officially become a Leeds player.
The game-changing transfer would represent a hugely significant one in the modern-day history of Leeds, preparing for their return to the Premier League after a 16-season absence and sporting director Victor Orta has professed the club's excitement at landing a 'world-class' player.
That sense of satisfaction is also shared by Marcelo Bielsa, according to Orta - with working alongside United's revered Argentine head coach known to be a key consideration in Rodrigo's decision to head to England.
Speaking ahead of the completion of the deal in Spain, Orta said: "He (Bielsa) is very happy. He wanted this leap of quality and when I put this name in the table, he took one minute or 30 seconds to ask if that was possible and here we are.
"We could make it and we hope we finalise it all as that is the most important.
"I have been in football for 16 years and this is one of the first times that I see the three parts very happy and this makes me very proud. We are all very happy and obviously we receive a world class player and we are very excited.
"It is true that this is the most expensive signing of Leeds's history.
"I would like to thank my president Andrea Radrizzani, and our director Angus Kinnear that believed in all of this. I would like to thank the player and the player's father that believed the project since the start and that allowed us to present what we are."