The UCU claimed the university had put more than 450 jobs at risk, including academics in archaeology and forensic science, biomedical sciences, psychology and public health.
It said the university was also looking to close its chemistry and film and television courses.
The strikes come after 67% of UCU members voted in favour of industrial action, the union said.
After Monday, further walk outs are scheduled for Monday 7, Tuesday 8, Thursday 10 and Friday 11 July. Staff will then strike for five consecutive days between 21 and 25 July.
Ms Grady said: “Bradford staff have no choice but to down tools because they refuse to see important courses cut, jobs axed, and staff and students pay the price for management’s financial failings.
“The vice-chancellor now needs to extend the consultation period and rule out compulsory redundancies.
“If she refuses to do so, there will be sustained disruption on campus.”
Responding to the announcement, a university spokesperson said it hoped to keep disruption to students to a minimum.
“We appreciate this is a difficult time for many at the university and we will continue to offer support wherever possible,” they said.
“Like many UK universities, we are facing significant financial challenges and addressing these remains a priority in order to safeguard the long-term future of the university.”
