Actually, before Bella and Edward's encounter, Tanya did attempt to flirt and possibly start a relationship with Edward. Back then, Edward politely rejected her feelings, proving he had no interest in her. After Bella arrived in Edward's life, Edward quickly relocated to Denali where Tanya and her coven mates lived, so that he wouldn't drink Bella's blood. Tanya takes this as a confirm to their relationship. Again, Edward rejects her.
Tanya and Edward never had a relationship, Edward had very little interest in Tanya. In the beginning of 'Twilight' Edward is shown to be very depressed and lonely, for he hasn't a mate. This proves that even though Tanya's interest in him, he refused it, rather wanting to be alone than with her.