Two separate petitions have collectively gotten over 18,000 signatures
The Taoiseach is being asked to boycott the annual St. Patrick's Day visit to the White House following President Donald Trump's 'anti-Muslim' executive order.
Speaking on RTÉ's This Week, Labour leader Brendan Howlin said if Mr Kenny cannot articulate the country's complete rejection of what President Trump stands for, then he should not go to the White House to celebrate St Patrick's Day.
However, he added he did not believe Mr Kenny would do that.
Meanwhile, Minister for Social Protection Leo Varadkar said the Taoiseach's St Patrick's Day visit to the White House cannot be just about smiles and shamrocks.
Speaking on RTÉ's The Week in Politics he said this county needs to work with the US in areas of common interest but it must also use its voice to raise concerns and stand by its values.
People Before Profit is putting forward a Dáil Motion this week to cancel the annual government visit
It's seeking political support across all parties with the hope Mr Kenny turns down the invitation to the shamrock ceremony on March 17th.
Richard Boyd Barrett said it would come as a real blow - even to Donald Trump.
"There is no doubt that the St Patrick's Day event have always been an important publicity opportunity for American presidents to ingratiate themselves with Irish America," he said. "So if an Irish Taoiseach refused to go it would be a significant blow."
In a statement, Minister for Foreign affairs Charlie Flanagan said he shares the views of his foreign counterparts on President Trump's order.
"While US immigration policy is a matter for the US authorities, it is clear that the most recent decisions could have far-reaching implications - both on humanitarian grounds and on relations between the US and the global Muslim community", he said. "Accordingly, I share the concerns of other EU partners regarding this most recent development."
Foreign Secretary Boris Johnson and Home Secretary Amber Rudd are to call their US counterparts to raise concerns about Donald Trump's travel ban.
They were given the order by Theresa May following global criticism of the US President's executive order on immigration.
Downing Street has said that the Prime Minister does "not agree" with the controversial measures.
Mr Johnson and Ms Rudd were told to speak out over the block on refugees and many Muslims.No 10 is particularly concerned about the impact the ban could have on UK nationals, including Iraqi-born Tory MP Nadhim Zahawi and Somali-born athlete Sir Mo Farah.