Lord Geidt resigned on Wednesday after he said he was put in an "impossible and odious" position by the PM.
But Mr Johnson has yet to commit to appointing a successor.
Labour's proposals, if passed, would enable a committee to appoint an independent ethics adviser if the role is unfulfilled for two months.
The motion will be put to the Commons on Tuesday, with Labour's deputy leader Angela Rayner saying the role needed to be "urgently reformed".
Lord Geidt's role was to advise the PM on the ministerial code - a set of rules governing standards of behaviour.
In announcing his resignation, he said he had been asked for advice on an issue he believed would amount to a deliberate breach of the code.
"This request has placed me in an impossible and odious position," wrote Lord Geidt in his resignation letter.
The former ethics chief also said he had come close to quitting over the Covid-law breaking in Downing Street.
Downing Street accepted that Lord Geidt fulfilled a "vitally important" role, but said the prime minister was reviewing the position and could abolish it.
Labour's proposals would give powers to the Commons Public Administration and Constitutional Affairs Select Committee to appoint an independent adviser on possible breaches of the code and initiate its own investigations.
The new "adviser on ministers' interests" would advise the committee on any potential violations and on the effectiveness of the ministerial code.
They would need to consider any possible breaches referred to them and could also conduct their own investigations, the Labour party said.
Ms Rayner said it was time for Conservative MPs to back the proposals to "clean up politics".
"Labour will put No 10 in special measures to prevent this prime minister running roughshod over the rules, dodging accountability and degrading standards in public life," she said.
Lord Geidt was the second ethics adviser to quit under Mr Johnson, after Sir Alex Allan left in 2020.