Ms Leadbeater told the Yorkshire Post: “I would love to represent this extraordinary, vibrant place that I have called home all my life."
She would be "honoured", she said, to "give the people of Batley and Spen the opportunity to put their trust in me" in the vote expected to take place this summer.
The constituency was held by her late sister until her murder in June 2016.
But Labour faces a huge test to cling on to the West Yorkshire seat, where it is defending a majority of just 3,525 votes.
The party’s defeat in its former seat of Hartlepool has thrown its popularity among northern voters into the spotlight.
Earlier this week the party’s new national campaign co-ordinator Shabana Mahmood acknowledged the potentially difficult by-election the party faces in Batley and Spen.
She told BBC Breakfast: “It is no doubt going to be a big test. It is an important by-election for us.
“We have just won in West Yorkshire and that is the base on which we have to build.
“I am in no doubt as to the amount of work that needs to be done.”
Ms Leadbeater, who is an ambassador for the Jo Cox Foundation, was awarded an MBE last December.
The vacancy was created after the local MP Tracy Brabin was elected as the first mayor of West Yorkshire on Sunday.
Ms Brabin, who played Tricia Armstrong in Coronation Street, became an MP in a by-election after the murder of Ms Cox.
In her victory speech Mr Brabin said politics was “better” when women were at the “top table”.
“Growing up on free school meals, in a Birstall council flat, inspired by the Batley Variety Club to pursue life as an actor, I never imagined I would be elected as a Member of Parliament in my home town, let alone be asked to serve as the first ever metro mayor of West Yorkshire. The first ever woman metro mayor in the country. Our politics are better when women are at the top table.”