Irish ministers have said they want the UK to stay in the EU and will not stay neutral in the forthcoming referendum.
The UK is the Irish Republic's largest trading partner, with one billion euros in weekly trade between the countries.
Dublin has begun contingency planning in case the UK chooses to leave.
Ahead of Mr Kenny's visit to Downing Street, Ireland's Public Expenditure and Reform Minister Brendan Howlin said there were serious concerns about the possibility of a UK exit and Ireland would express its views clearly on the referendum, which will take place some time before the end of 2017.
"I think probably for the first time ever it will be our business because it will have a direct and significant impact on Ireland and the Irish people and we certainly will be expressing our views on that," he said.
"By tradition we don't involve ourselves in domestic issues - for example in the Scottish referendum the Irish government was entirely silent - but on this matter I don't think the Irish government or indeed the Irish Parliament will be silent."
Europe minister Dara Murphy said the government's current strategy was to try to help David Cameron in his efforts to change the terms of the UK's EU membership.
He would not be drawn on Ireland's attitude to Mr Cameron's key demands, saying he had yet to see specific proposals on the table, but he told the BBC that his government had "some concerns".
"I think the best way of dealing with our concern is for our Taoiseach Enda Kenny meeting with David Cameron, by me and others engaging with our opposite numbers, and by talking to people in the UK and saying, 'the European Union is imperfect but we're far better off with it than without it'.
"It's in nobody's interest to step outside."
Former Prime Minister Bertie Ahern was a key figure in the Northern Ireland peace process, working alongside Tony Blair to help bring about the Good Friday Agreement.
He says it would be "senseless" for the UK to leave the EU and it would have profound implications for Northern Ireland, jeopardising much of the economic investment so vital to the efforts to resolve outstanding differences.
He said: "You can just imagine the difficulties caused if we were to go 100 kilometres up the road and you were to find a region that was not part of the European Union. The farmers in the North very much are supportive of the Common Agricultural Policy; merchandise, goods and services from the North are mainly into Europe. It would be senseless.
"And, in fact, I think it would probably be the only time we'd get all the parties in the North voting the same way. It would be going back light years, you would be going back to old times, which would be hugely negative."
Many business are worried too.
Sean O'Driscoll, the chief executive of Glen Dimplex, says the UK referendum is already creating uncertainty at a time when his business, which claims to be the biggest heating manufacturer in the world, needs to make key decisions about its future operations.
David Cameron is starting renegotiation of the terms of Britain's EU membership ahead of a referendum. Here is some further reading on what it all means:
Q&A: The UK's planned EU referendum
UK and the EU: Better off out or in?
What Britain wants from Europe
Timeline: EU referendum debate
About 30% of the Dublin-based company's turnover is in the UK, where it employs 3,500 people.
He told me it would be "catastrophic" if the UK was to leave the EU and it would have serious implications for his business, its employees, suppliers and customers.
Economists estimate that a British exit from Europe would cost Ireland anything from 2% to 12% of its GDP, depending on the terms of any future arrangements.
Brian Lucey, professor of finance at Trinity College, said it would be the greatest upheaval for the Irish economy short of a war.
But he said there were also potential benefits for Ireland in attracting some of the investment and financial services which are currently drawn to London.
It's a scenario the Irish government hopes to avoid, stressing its determination to help the UK to stay in the EU.
But when the Irish PM Enda Kenny arrives in London, David Cameron will want to know whether he can count on Ireland's support on the key changes he is seeking to Britain's relationship with the EU.
David Cameron will meet the Irish PM Enda Kenny on Thursday as he continues his talks with other European leaders to try to win support for changing Britain's relationship with the EU.