The London Evening Standard says Spurs and the Blues are both desperate to sign a striker, but time is running out before Friday's 11pm deadline.
Harry Kane's absence until at least April following hamstring surgery has pushed Tottenham into the transfer market, with Giroud emerging as a surprise option as he enters the final six months of his Stamford Bridge contract.
Chelsea, however, are reluctant to allow the Frenchman to leave without finding a replacement and their ongoing struggles in doing so have left the 33-year-old in limbo.
Inter Milan are thought to have agreed a deal in principle with Giroud, with the Blues willing to accept around £5million, but the situation has dragged on and Inter's interest is thought to have waned, with reports in Italy this morning claiming they would prefer now to wait until the summer.
That could present an opportunity for Spurs, who are believed to have indicated they will offer an 18-month deal.