One survey this week recorded support for continued European Union membership at 53 per cent.
While there she also addressed the issue of the border between Northern Ireland and the Republic of Ireland, one of the most crucial and contested issues to come out of Brexit.
The companies surveyed - the biggest or most internationally-focused banks, insurers, asset managers, private equity firms and exchanges in Britain - were responding to questions about their plans in the event of a so-called "hard" Brexit, where the United Kingdom would leave not only the EU but also the single market and Customs Union.
At this midpoint in the Brexit process, May - weakened by a failed snap election in June a year ago and who has been undermined by hardline Brexiteers within her own cabinet - can however count on only an ambivalent public opinion.
Shadow chancellor John McDonnell defended Ms Thornberry's comments, emphasising her "sarcasm" in response to a journalist's question.
On Wednesday, shadow foreign secretary Emily Thornberry faced criticism from within her own party for saying Labour MPs would probably approve the government's "blah, blah, blah" deal that would pass Labour's six tests - which included maintaining the benefits of the single market and customs union.
At some stage - most likely in October - the PM will put an outline of the Brexit deal to Parliament.More news: Weather: Drop in temperature, scattered rain expected
Mr Blair, a strong backer of United Kingdom membership of the European Union, told BBC Radio 4's Today the "sensible" option was to "take a final decision" once the terms of the deal have been set out.
Scottish National Party spokesman Michael Russell accused Mrs May of planning a "power grab" by repatriating some responsibilities now exercised in Brussels to Westminster rather than Edinburgh.
'By pursuing a disastrous hard Brexit, regardless of the cost to jobs and living standards, Theresa May and the Tories have shown they think they can now do anything to Scotland and get away with it, ' said Mr Russell.
Speaking during her visit to Alex Begg weavers in Ayr, which produces luxury cashmere scarfs and accessories, Mrs May was asked about how the Nationalist leadership was claiming she personally was the hindrance to a breakthrough on the bill.
"Brexit provides us with opportunities".
On Thursday, May will visit a textile factory in Ayrshire, a parent and toddler group in Newcastle, have lunch with farmers near Belfast before meeting businessmen in Barry, south Wales.
"I am determined that as we leave the EU, and in the years ahead, we will strengthen the bonds that unite us, because ours is the world's most successful union".
She will then go on to Barry in south Wales for a round-table discussion with businesses, before completing her tour in west London with a group of Polish citizens who have made the United Kingdom their home.