The Absent Voting (Elections in Scotland and Wales) Bill passed its second reading on Friday without a division and will undergo further scrutiny at a later date.
Voters throughout Britain can apply for a postal ballot paper online for general elections and national referendums.
But residents of Scotland have to use paper forms if they would like a postal vote for elections to the Scottish Parliament or their local councils.
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Residents in Wales also have to use paper forms for Senedd elections.
If it clears both Houses of Parliament, the Bill would change this system to bring electors in line throughout Britain.
Democracy minister Rushanara Ali described the paper form process as one which “could be cumbersome and difficult as a process and one that some electors may have had to rely on others to support them”.
She added the UK Government is “wholeheartedly in support of the aim of removing that divergence” in voting practices throughout Britain.
Labour MP for Edinburgh North and Leith Tracey Gilbert, who brought forward the proposals as a private member’s bill, said: “If an elector in Scotland or Wales wishes to have a postal or proxy vote for a devolved parliament or local election, they are still required to fill out a paper application form and physically sent it in to be processed.
“This Bill seeks to end this inconsistency and give voters in Scotland and Wales equal choice in how they apply for an absent vote for use in the Scottish Parliament, Senedd Cymru and local elections.”
She added: “It’s critical that we make voting as accessible for everyone, and particularly for young people and for disabled people, to encourage as many people as we can to vote.”