The 29-year-old, who beat Novak Djokovic in the ATP World Tour Finals on Sunday, has been tipped to be named in the New Year honours list.
However, the Wimbledon champion and successive Olympic gold winner said he could "still mess up" and "make mistakes" in his career.
Murray was previously honoured with an OBE after winning the US Open in 2012.
In order to receive an honour, the names of nominees must be put forward by members of the public or government departments before being considered by a selection committee.
Their assessments are then passed to the Queen through the prime minister, who then informally approves the honours.
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Murray became tennis world number one at the beginning of November after a year which saw him reach 12 finals in 13 events and achieve an impressive 24 wins in a row.
Among those calling for him to be recognised are businessman and TV personality Lord Sugar, while former prime minister David Cameron said back in 2013 that he could not think of anyone "more deserving" than Murray of a knighthood.
Speaking about the possibility, Andy Murray said: "Obviously it is the highest honour you can get in this country. I don't know, I feel too young for something like that.
"I don't think about that stuff much, really. When I win any award or am presented with anything it is nice because it is recognition for what you have given your life to, up to now anyway.
"I am still young and there are still a lot of things that can go wrong, I could still mess up and make mistakes. Do stuff wrong. I am just trying to keep doing what I am doing, working hard, and achieving stuff."
While a knighthood remains a possibility, Murray is now favourite to be named this year's BBC Sports Personality of the Year - an accolade he has already won twice.
However, he said his focus now was the impending wedding of his father, which would make it difficult for him to attend this year's ceremony.
He added: "I don't think I am going to be able to go this year. I've got my father's stag do next weekend, then he is getting married the following weekend.
"So I will go up to Scotland on Saturday, come back down to London for four or five days then come back up to Scotland. Then I go to Miami the day after my dad's wedding."
Tennis world number one Andy Murray has said he is "too young" for a knighthood.