The deeply conservative state, home to the Christian Social Union, the more straight-laced sister party to Angela Merkel's Christian Democratic Union, seems set to give the Left party five percent of the vote, enough to send a deputy to the parliament.
The poll, set to be published in the Bild am Sonntag paper on Sunday, also suggests the CSU may have difficulty defending its absolute majority, losing around nine percent from the state election in 2003, to reach 51 percent.
The CSU is holding its party conference in Nuremberg this weekend, to decide its manifesto for the election campaign, with issues such as tax reform and the return of the commuter tax break high on its list.
These topics have provoked arguments with the CDU, with Merkel rejecting the proposed commuter tax break out of hand.
She addressed the CSU meeting, showering them with praise, saying that Bavaria could serve as a model for the rest of the country, and thus defusing the row which had threatened to run out of control. The CSU was forced to focus its frustration in attacks on the Social Democrats, which in Bavaria seem doomed to bump along with around 19 percent of the vote as they did in 2003.
A Left party deputy in the Bavarian state parliament would be a massive symbolic victory for the socialists, striking deep into established German political territory and widening its geographical reach.
Until now the party, which unites the successors of the eastern German communists with western socialist rebels, has performed best in the east, but has achieved success recently in western regions of the country.
A further poll showed only around half of Germans believe the country is well-off, a dramatic drop in optimism compared to two years ago when there were actually 1.5 million more people without work.
The poll, conducted for the Welt online newspaper, also showed that nearly 60 percent of people think the standard of living will drop in the future, while the demand has grown for greater state intervention to secure a certain level of individual prosperity.