The 23-year-old, who is currently on holiday, has one year remaining on his contract at Tannadice and had told the club he will not sign an extension.
He will reportedly undergo a medical at Derby's training ground on Monday.
"Dundee United can confirm that an undisclosed fee has been agreed with Derby County for Johnny Russell," said a statement on the SPL club's website.
"The transfer is expected to be concluded after the player has completed the necessary formalities early next week."
A Derby spokesman added: "We can confirm that a bid for Johnny Russell has been accepted.
"However, the club will be making no further comment until a deal is done."
Russell, Dundee United's top scorer last season with 20 goals in 38 games, missed two months of the campaign due to a broken leg, but returned for the final three matches.
The former Scotland U21 international has netted 46 goals in 120 appearances for the Tangerines.