(Adds details on investigation, currency reaction)
SAO PAULO, July 16 (Reuters) - Federal prosecutors in Brazil have opened a formal inquiry into whether former President Luiz Inácio Lula da Silva improperly used his connections overseas to benefit Latin America's largest engineering firm, Odebrecht SA, a spokeswoman said on Thursday.
In May, prosecutors in Brasilia said they had opened a preliminary investigation into alleged influence peddling, saying the former leader had frequently traveled abroad at Odebrecht's expense after leaving office, from 2011 until 2014.
The inquiry puts the legacy of one of Brazil's most popular former leaders on the line at a time when some are calling for the impeachment of his chosen successor, President Dilma Rousseff, for alleged campaign finance irregularities.
A spokeswoman for Lula's institute, the Instituto Lula, said the institute was surprised by the news, and thought the inquiry had been escalated too quickly. She said Lula's travels were completely legal.
Brazil's currency, the real, weakened after reports of the investigation, closing 0.75 percent lower at 3.153 per dollar.
In their preliminary inquiry, prosecutors cited media reports that Brazil-based Odebrecht had won contracts in countries including the Dominican Republic and Cuba after Lula met with their leaders, traveling at Odebrecht's expense.
Lula may have also improperly influenced Luciano Coutinho, president of state-run development bank BNDES when it approved loans for Odebrecht projects abroad, prosecutors said.
News of the full-blown investigation of Lula comes a month after Odebrecht's chief executive officer, Marcelo Odebrecht, was arrested. Prosecutors in the southern city of Curitiba say his family-run conglomerate may have helped lead what they call a cartel of engineering firms accused of fixing prices and overcharging state-run oil firm Petrobras.
Neither Lula nor Rousseff are being investigated in the scandal known as "Operation Car Wash" but Rousseff's approval rating has suffered as a result of the probe.
Plea bargain testimony from engineering executive Ricardo Pessoa, obtained by Veja magazine last month, said money originating from overpriced Petrobras contracts had helped finance her 2014 re-election campaign.
Rousseff and the Workers' Party say all campaign donations were legal and Rousseff has urged a thorough investigation into graft at Petroleo Brasileiro SA, as the oil major is formally known. (Reporting by Caroline Stauffer, Tatiana Ramil and Eduardo Simoes; Editing by Peter Galloway and Andrew Hay)