Britons are due to vote in European Parliament elections on May 23, putting Theresa May's Conservative Party in a tricky position. 

The prime minister doesn't want the polls to happen, so the party hasn't launched a campaign. But the only way to cancel them is to pass a Brexit deal through the UK parliament before voting day. 

To do that a breakthrough in the ongoing cross-party talks is needed. And for the Conservatives and the opposition Labour Party to reach agreement probably requires the government to accept a permanent customs union with the EU, an option it has previously ruled out and many of May's party strongly oppose.

May reportedly considered putting key Brexit legislation – the withdrawal implementation bill – to MPs this week in a bid to break the impasse. But she is unlikely to risk the move without a signal of Labour support, and there is little incentive for the opposition party to do so with local elections set to take place on Thursday.

The Conservatives are expected to suffer a drubbing in the elections, and Labour to make gains. Local elections expert Robert Hayward, who sits for the Conservatives in the House of Lords, predicts that his party will lose 800 councilors and Labour gain 300.

That would be the Conservatives' worst result for nearly 25 years and renew pressure on May's position. And, if the European elections do go ahead three weeks later, an even bigger defeat for the prime minister's party is expected to follow.

An Opinium survey on Sunday placed the Conservatives on just 14 percent, with Labour and the new Brexit Party tied in the lead with 28 percent. Other polls put the Brexit Party, headed by veteran anti-EU campaigner Nigel Farage, clear in the lead. It is taking votes from the Conservatives with a simple and focused message: Leave the European Union.  

All of which piles pressure on May to find a solution. 

Michel Barnier, the EU's chief Brexit negotiator, said on Sunday that he expects the results of the cross-party talks this week. But Labour's business spokesperson Rebecca Long-Bailey warned that the government has not moved "on any of their red lines," such as a customs union.  

There is no good option for the prime minister. If she were to accept the permanent customs union demand, there's a chance a Brexit deal would be passed in time to avoid the European elections.

But a customs union would be seen as a betrayal by many Conservative MPs, who believe it would prevent an independent post-Brexit trade policy. A split in the party would be cemented and a fresh attempt to oust May would likely follow. 

And in any case, if Labour does agree to a deal it is likely to hold off on signing up until after the elections to maximize Conservative discomfort.

If May continues to resist a customs union, two bad elections with no progress on Brexit would also lead to fresh leadership speculation.

Whatever happens, May is set to be a testing month for the British prime minister.