A video has emerged of Jones making the remarks at an event for Japanese company Fuso last July.
The Australian said: "We've played 23 Tests, we've only lost one Test to the scummy Irish.
"I am still dirty about that game. We'll get that back, we'll get that back don't worry. We've got them next year at home, we'll get them back."
Jones also reflected on his time as Japan head coach, making reference to a 125-0 defeat for the Under-20 national team against their Wales counterparts soon after he took over the Brave Blossoms.
"Who knows Wales? Are there any Welsh people here? So it's this little s*** place that has got three million people. Three million!" he said.
Jones on Wednesday expressed his remorse for the comments.
"I apologise unreservedly for any offence caused - no excuses and I shouldn't have said what I did. I'm very sorry," he said in a statement.
A Rugby Football Union spokesperson said: "Eddie has apologised for his inappropriate remarks, and the RFU is also very sorry for any offence caused. We will be apologising to the IRFU and WRU."
England will attempt to deny Ireland a Grand Slam at Twickenham on Saturday.