The 38-year-old is assistant to boss McCarthy and will remain in his international role following his appointment onto Middlesbrough's coaching staff under former United defender Jonathan Woodgate.
Bamford's international allegiance has been a hot topic of debate with the Leeds United striker available to represent Ireland through his grandparents.
McCarthy has been frustrated in his attempts to lure the 25-year-old to represent his side and revealed last week that he would no longer chase the forward leaving it up to him to "pick up the phone."
Keane, who was speaking to The Irish Sun, echoed his manager's thoughts, stating: “I spoke to you about it years ago, do you remember? And the values are still the same, they will never change.
“You’re representing your country and if you want to play for Ireland, you come play for us. We’re not going to go and chase you.
“I’ll tell you with Patrick Bamford, he should be chasing us. He should want to come here. If he doesn’t want to, no problem.
“We shouldn’t be chasing people, by the way. That’s what f***ing winds me up. We shouldn’t be chasing people.
“They should want to play for us. If they don’t, then no problem. Why should we go and chase people? We are f***ing Ireland.
“We have qualified in the last few years for two Euros, World Cup 2002. If you want to come, come. Represent your country and be proud to play for your country.
“If you don’t, don’t waste anybody’s time. Don’t waste the manager’s time.
“And I am not speaking specifically about Bamford. I am talking about everybody. Anybody.
“Don’t just use us as a tool, if you like, for maybe other countries to come in for you. Be proud of your country. Be proud to play for Ireland."