STOCKHOLM, June 21 (Reuters) - Prime Minister Stefan Lofven looked destined to be the first Swedish premier to be ousted by parliament on Monday after his government's weekend efforts to broker a deal to win back the support of the Left Party fell through.

That would leave Lofven with one week to either resign or call snap elections, unseen in Sweden since 1958. In either case, national elections set for September next year would proceed as planned.

"It is looking like we will have a historic day in parliament," national daily Dagens Nyheter said on its front page.

Lofven secured a second term in 2018 only after months of negotiations following an election in which the anti-immigration Sweden Democrats made big gains to redraw the political map.

A former union boss known for his negotiating skills, Lofven has since led a fragile minority government of Social Democrats and Greens, supported by former political rivals the Centre Party and the Liberals.

While not part of any formal agreement, the government has also needed the tacit backing of the Left Party. That changed last week after the government signaled it would move to ease rent control for new-build apartments, a red flag for the Left.

The formerly communist Left Party demanded that the government withdraw any such plans.

While the Social Democrats and the party's partners took steps to placate the Left over the weekend, the last-gasp moves appeared to have failed to stave off the political crisis.

The motion of no confidence called by the fiercely nationalist Sweden Democrats, set to be voted by parliament at about 0800 GMT, now looks all but certain to gather the votes needed to oust Lofven.

Economists have said they do not expect the political uncertainty to weigh on the economy, given the strict fiscal rules under which the country operates. (Reporting by Niklas Pollard and Johan Ahlander; Editing by Clarence Fernandez)