Cox was reportedly shot and stabbed in Birstall near the northern city of Leeds and has been airlifted to Leeds General Infirmary in a critical condition. Eyewitnesses have suggested the shooter was an older man in his 60s or 70s.
West Yorkshire Police said in a statement: "At 12.53 today, police were called to a report of an incident on Market Street, Birstall, where a woman in her 40s had suffered serious injuries and is in a critical condition.
"A man in his late 40s to early 50s nearby also suffered slight injuries.
"Armed officers attended and a 52-year-old man was arrested in the area. There are no further details at present. Police presence in the area has been increased as a reassurance to the community."
The 41-year-old is Labour MP for Batley and Spen. She has been head of policy for Oxfam, national chair of the Labour Women's Network, a senior advisor to the Freedom Fund, an anti-slavery charity, and is currently Chair of the Friends of Syria All-Party Parliamentary Group.