She said it was time to "break the logjam" with more peers attending the Lords than ever before.
There are currently 781 peers eligible to take part in Lords proceedings.
A new list of peerages is expected next week.
The Dissolution Honours List is made following the end of every Parliament.
Ms Cooper, the shadow home secretary, said Lords reform was "long overdue" and accused Prime Minister David Cameron of "trying to flood the House of Lords with more Tory appointments".
She cited LSE research which found that 62% of Lords appointments since 2010 had been to the government benches - more than under other recent prime ministers.
Members of the House of Lords are appointed by the Queen on the advice of the prime minister.
Attempts to reform the system were abandoned in the last Parliament amid strong backbench opposition, particularly among Conservative MPs.
Another Labour leadership contender, Andy Burnham, says he would introduce an "indirect election" system based on votes cast at the general election, which would create a "proportional Lords".
Liz Kendall and Jeremy Corbyn have also argued for the Lords to be elected.
The Conservatives, who have a 12-seat majority in the House of Commons, have 224 peers, compared to Labour's 214 and the Lib Dems' 101. There are also 178 crossbench peers.
The SNP says it will continue to refuse to sit in the "bloated, ridiculous, unelected second chamber".
Ms Cooper also criticised the government for the way it is introducing "English votes for English laws", replacing the Human Rights Act and changing the rules on party funding.
She called for a new "convention" to advise on constitutional changes, including Lords reform, and said that as leader she would "go ahead anyway" if Mr Cameron rejected her plans.
There should be a freeze on the appointment of new peers until the House of Lords has been reformed, Labour leadership contender Yvette Cooper says.