UPDATE 2-Sterling dips as Brexit concerns scupper rally

* Graphic: World FX rates in 2018 http://tmsnrt.rs/2egbfVh

* Graphic: Trade-weighted sterling since Brexit vote http://tmsnrt.rs/2hwV9Hv (Adds quote, context, updates figures)

LONDON, June 4 (Reuters) - Sterling fell on Monday as concerns about Brexit clouded the outlook for the currency after a brief rally encouraged by stronger-than-expected data on Britain's construction sector.

The pound fell 3.43 percent versus the dollar in May, its biggest monthly decline since 2016, as weakness in the UK economy and non-UK factors including new U.S. trade tariffs impacted the currency.

Sterling started June on the front foot, buoyed by data showing signs of a possible strengthening of the British economy after a sluggish first quarter.

Early in Monday's session, sterling hit a seven-day high of $1.3398, amid broad dollar weakness and a purchasing managers' index (PMI) for Britain's construction sector that showed activity at 52.5 for May, higher than 52 predicted by analysts.

But the pound later relinquished all of its gains and fell 0.3 percent to $1.3305. It also slid half a percent against the euro to 87.85 pence as the single currency rallied on receding worries about a political crisis in Italy.

Analysts said the pound's decline was partly due to a flurry of headlines about Brexit and Britain's future EU ties.

"Scepticism that Downing Street will have proposals in place by an EU summit in June weighed on sterling, amid a weekend of doomsday scenario reports," said City Index analyst Ken Odeluga.

With time running out for Britain to secure a deal before exiting the European Union next March, diplomats are hoping an EU summit on June 28-29 could break the deadlock.

The legislation underpinning Britain's exit from the European Union will return to the House of Commons on June 12, The Times newspaper reported on Monday, giving lawmakers the chance to discuss proposed changes to the bill which could shape Brexit.

The EU says London is yet to come up with a viable solution for the Irish border - the fear is that reinstating a physical border, including for customs, between EU-member Ireland and Britain's Northern Ireland province could revive violence there.

Investors are also studying economic data for signs of whether the Bank of England will raise interest rates to curb inflation this year.

Bad weather and slow growth saw markets in May drastically scale back expectations for monetary tightening by the BoE.

"If the economy gains some momentum in June that will boost the chance of a BoE rate hike in 2018 and fuel further demand for the pound this month," WorldFirst head of FX strategy Jeremy Cook said.

Data on Tuesday is also expected to show some month-on-month growth in the UK's vital services sector. (Reporting by Tom Finn Editing by Gareth Jones and Andrew Heavens)