According to reports in Spain, Ramos has agreed a four-year deal and is expected to join Spurs on Monday.
The Evening Standard newspaper reported Ramos would be paid £25m over the length of his contract.
"This has been the hardest decision of my life," added Ramos. "My time at Sevilla has been the most important period of my life personally and professionally. I will be eternally grateful."
Sevilla's physical trainer Marcos Alvarez has also tendered his resignation.
Ramos' contract with Sevilla had been due to expire at the end of this season.
"The now ex-Sevilla coach declined the possibility of giving a press conference at the club's training facilities," said a Sevilla statement.
"He has chosen his website www.juanderamos.com as a means of saying goodbye to Sevilla."
