How important are credit reports for renters?

Is there any relief or consideration given for homeowners whose credit has been ruined by home ownership? We currently have a very bad mortgage and our bank is unwilling to modify (not a Fannie Mae or Freddie Mac and HSBC Mortgage Services, so not obligated to honor any of the government programs) and it looks like we are headed towards foreclosure. If we can prove that we have a steady, professional income and the rental payment that we are seeking is 1/2 the mortgage that we were able to pay, just at the expense of everything else, is the realtor allowed to take those factors into consideration? Please advise Thanks