The clubs have reportedly agreed a fee of £7 million with Southampton midfielder Richard Chaplow possibly returning to Turf Moor in exchange.
Rodriguez finished the season with 21 goals in all competitions and earned a place in the PFA Championship Team of the Year.
He made his debut for the Clarets in 2007 and was recently linked with a move to fellow Championship club Cardiff City, but it now seems like he could be heading for the Premier League.
Source: DSG