In The Great Gatsby, do Daisy and Tom Buchanan relish a great reputation?
I would say that Tom and Daisy relish their lavish lifestyle. They live on an enormous, beautiful Long Island estate with an expansive front lawn that runs all the way down to the waters of Long Island Sound. The house and grounds are very impressive to Nick when he first sees Tom and Daisy's home. As for their reputations, Tom and Daisy are members of an elite social class which gives them a certain prominence. When he was young, Tom enjoyed a reputation as a football player for Yale University. Nick said Tom was "a national figure in a way." Daisy, as a girl in Louisville, was well known in her own elevated social circle. Tom and Daisy both grew up with money and the prestige it gave them.
As an adult, Tom doesn't seem especially concerned about his reputation. He has had numerous affairs outside his marriage, and was once involved in a car wreck that got his name in the papers, along with the name of the hotel maid he was with. He does attempt to keep his affair with Myrtle quiet, however. He keeps an apartment in New York for their meetings and does not let Myrtle's husband know of his involvement with Myrtle. Tom's money insulates him to a certain extent. Daisy flirts shamelessly with Gatsby at one of his parties and disappears with him for a long period of time, which Tom notes. Daisy's actions indicate that she is not overly concerned with her reputation. Again, money seems to give her the confidence to behave as she chooses. After Gatsby is murdered by George, Tom and Daisy leave the country quickly, probably to avoid gossip and rumors as well as legal inquiries into their involvement in the matter.