Mohammed Abbas, 31, is wanted in connection with three assaults which left the victim - a woman in her 30s - with "significant injuries".
The incidents occurred on June 9 and 10 at an address in Nelson, according to police.
Insp Carole Langhorn, of Burnley CID, said: "We would like to locate Abbas as part of our enquiries into three assaults.
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"We are now urging anybody who has seen him, or knows where he may be, to get in touch with us as soon as possible."
Abbas, from Nelson, has been described as Asian, around 5ft 6in tall, with short black hair and brown eyes.
He has links to Blackburn and Nelson.
Anybody with information has been urged to contact police by calling 101, quoting log number 1425 of June 11.