The German telecommunications company and the British mobile group are both looking to buy 20 to 25 per cent of Swisscom's mobile phone division, according to the German magazine "Der Spiegel."
Vodafone has reportedly made the better offer so far.
A spokesman for Swisscom, Christian Neuhaus, said the company has held talks with several bidders but he refused to comment on industry rumours. There has been speculation that a decision could come by the end of the month.
A stake in Swisscom Mobile would allow either Vodafone or Deutsche Telekom to participate in the next bidding round for third generation mobile telecommunications licences (UMTS), which will take place on November 13.
The successful bidder would be able to acquire even more shares in Swisscom Mobile if the Swiss government decides to reduce its own shareholdings. It currently holds a 65.5 per cent stake in Swisscom.
swissinfo with agencies