* German results due on same day Italy holds election
* Polls show majority of SPD supporters back coalition
* Tabloid causes stir with report of dog as SPD member (Adds comment from SPD on dog)
BERLIN, Feb 20 (Reuters) - Members of Germany's Social Democrats (SPD) began voting on Tuesday on whether to enter a new coalition with Angela Merkel's conservatives, a postal ballot which could scupper the chancellor's chances of a fourth term in office.
If the SPD's nearly half a million members reject the deal, a new election or a minority government in Europe's biggest economy is likely. Either would be a first for post-war Germany, now without a formal government for nearly five months.
The result of the vote, which runs to March 2, is wide open and will be announced on March 4. That will be the same day Italy goes to the polls in a vote seen as too tough to call, as European politics splinter after years of austerity and waves of migrant arrivals from war-torn Syria and elsewhere.
Polls indicate a majority of SPD supporters back the deal with the Christian Democrats (CDU) but in the absence of any internal surveys it is unclear whether its members feel the same way.
The SPD's leaders want the coalition to go ahead, but there is also strong opposition within the party grassroots.
The party has had a chaotic few weeks, plumbing record lows in opinion polls and sinking into a messy leadership change after a personal row between former leader Martin Schulz and Foreign Minister Sigmar Gabriel.
Opponents of the deal blame their party's role in a "grand coalition" with Merkel from 2013 to 2017 for their worst performance since World War Two in the Sept. 24 election.
In January, a narrow majority of SPD delegates at a special congress endorsed entering formal coalition talks with Merkel. On Sunday, a poll showed that leading SPD mayors also favoured joining a coalition.
SPD leaders - Olaf Scholz, the interim party chief, and Andrea Nahles, who is expected to take over in April, are touring the country to convince members to approve the deal.
They are arguing that the grand coalition deal will help improve the lives of families, workers, pensioners and students and have stressed how they forced Merkel to concede the powerful finance ministry to the SPD and also leave the prestigious foreign affairs portfolio to the centre-left party.
"I am confident that the strong content of the agreement will convince the majority of members," Manuela Schwesig, one of the SPD's deputy leaders, told the Rheinische Post daily.
However, the substance of the coalition deal falls short for many in important areas including health and migrant policy.
The vibrant Jusos youth wing of the party and left-wingers are on their own tour, fighting for a "No Grand Coalition" vote and their message has resonated with many.
A campaign to get people to join the SPD and reject a coalition boosted membership by about 24,000 to almost 464,000.
The top-selling daily Bild caused a stir with a report that a dog from Spain called Lima had received an SPD card after an application was apparently made on its behalf to become a party member on Feb. 6. Personal details on the application included her gender and put her occupation as unemployed and her age at 21, in dog years, according to Bild.
All of this, said Bild, raised questions about possible manipulation.
Nahles responded by saying: "A dog can't vote." She added that the party would look into legal steps if there had been deception over a party member's identity.
It is the second time SPD members have voted on a coalition deal with Merkel. In 2013 almost 76 percent backed a tie-up.
Highlighting the depth of the SPD's woes, an INSA poll on Monday showed its support had slumped to 15.5 percent, below the far-right Alternative for Germany (AfD) for the first time.
Some analysts say the prospect of getting decimated in a new election may galvanise some SPD members to back the deal.
"Being seen as weaker than the AfD ... could well help backers of a grand coalition," said Oskar Niedermayer, political researcher at Berlin's Free University.
Immediately after the election Schulz promised to take the SPD into opposition, but Merkel's failure to clinch a coalition with two smaller parties forced him to reconsider. (Additional reporting by Thorsten Severin; Editing by Mark Heinrich)