text: Both Jenny Rathbone and Mike Hedges are applying separately to take part in a ballot which, if they win, will allow them to propose a bill to ban the fees.
  Mr Hedges said the fees were a "tax" on some of society's poorest with students among the biggest hit.
  But one letting agent said a ban could hit efforts to professionalise the sector.
  The Welsh Government said it was happy to consider how such legislation might work.
  The charges have already been banned in Scotland, while the UK government is planning to consult on doing the same in England.
  There are currently no firm plans to do the same in Wales, although the Welsh Government has said it was "actively considering" the move. Ministers have faced criticism for not acting sooner.
  AMs from all of the assembly's opposition parties have supported a ban.
  Assembly members can propose bills - proposals for new laws which when passed by the assembly become acts - from outside the government through a random ballot process.
  Mr Hedges said: "They are a tax on some of the poorest people in society who are engaged in private rented accommodation."
  The Swansea East AM said it was a big problem in university cities and a "big problem anywhere where there is more demand for housing then there is housing".
  Mr Hedges believes that about 20 AMs may apply to take part in the ballot - although only Mr Hedges and Ms Rathbone are currently known to be proposing banning the fees.
  Cardiff Council AM Jenny Rathbone said Mr Hedges' attempt "doubles the chances of one of us being picked".
  Catherine Iannucci, 21, originally from Barry and studying her third year of history and politics at Cardiff University, lives in a house with six other tenants.
  She said they have had to pay Â£80 each to take the home off the market - amounting to a fee of Â£560 for the agency.
  Ms Iannucci, who is a part of the group Labour Students and has conducted research on agency fees for Ms Rathbone, said: "It's a significant amount of money.
  "We don't really know what they're for and we don't know what that's gone towards."
  She also had to pay a deposit of a month's rent, plus Â£50.
  "In the middle of term, because we look around February time - nobody has that money to just throw away easily. It does put you under a lot pressure for the rest of the term," she said.
  She said she would be in favour of a ban as long as it meant that rents did not go up as a result.
  Douglas Haig is the director for Wales and vice-chairman of the Residential Landlords Association - which also owns Cardiff property management company James Douglas Sales and Lettings.
  He said his firm does charge fees, adding: "There are fees that go to landlord, there are fees that go to the tenant."
  Mr Haig said he did not condone "sharp practices" in the sector but warned that banning fees may make it harder to professionalise the industry.
  "If we just outright ban fees, then we're going to have a number of consequences," he said.
  "The most important is around the professionalisation of the sector... If you are taking away money from the sector, that's going to be very difficult to continue providing quality.
  "I don't think the vast majority of agents out there are actually profiteering.
  "There are costs in running these businesses that people are unaware of and as increased state regulation comes in, those costs continue to rise."
  A Welsh Government spokesman said: "We are happy to consider how legislation on this might work.
  "We want to look at the evidence from Scotland and see wider consultation to ensure that a ban on fees does not push rental costs up."
  The next scheduled ballot for backbench bills is on 25 January.
target: Letting agents' fees for tenants could be banned in Wales under proposals from two backbench Labour AMs.