Ahmed Farrah, aged 29, is wanted for questioning by detectives investigating the fatal stabbing of 21-year-old Kavan Brissett last month.
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Kavan was knifed in his chest in an alleyway close to Langsett Walk, near Walkley, on Tuesday, August 14.
He was rushed to hospital and underwent emergency surgery but died four days later when his life support machine was switched off.
Detective Chief Inspector Jude Ashmore said Farrah, who is also known as Reggie, '˜knows he is wanted' and is '˜deliberately avoiding' the police.
She has warned that anyone helping him evade arrest faces prosecution.
Anyone who spots Farrah should call 999.
Those with information on where he might be should call South Yorkshire Police on 101 and quote incident 827 of August 14.
The police incident room can be called direct on 01709 443507 or information can be passed to Crimestoppers, anonymously, on 0800 555111.