David Cameron will this week hand a peerage to the businesswoman Karren Brady, who became a household name as Sir Alan Sugar's aide on The Apprentice.

Brady, who introduced George Osborne at last year's Conservative conference, will be given a title as speculation circulates that the party could put her forward as a possible candidate to be London mayor.

Despite her imminent entry to the House of Lords, Brady ruled out a political career last year, saying: "I'm not going to become an MP and I'm not going to be London mayor. I have no political ambitions."

The 45-year-old started her career at Saatchi & Saatchi, before going on to work for David Sullivan, a multimillionaire with a pornography, football and property empire. After persuading him to let her run Birmingham City FC, she is now vice-chair of West Ham, which he owns with his business partner David Gold.

It is understood that the former Marks & Spencer boss Sir Stuart Rose is also being considered for a peerage.

Other possibilities include the Conservative co-treasurers Sir Michael Farmer and James Lupton, both of whom are leading party donors. Their inclusion on the list would almost certainly trigger a cash-for-peerages row, as the latest in a long line of party donors to enter the Lords.

There will be around 20 new peers, mostly selected by Cameron and the deputy prime minister, Nick Clegg, taking the numbers in the upper house to almost 800. The Labour leader, Ed Miliband, is rumoured to have nominated Michael Cashman, the former EastEnders actor and Midlands MEP.