Tutton added: “It’s my dream to go to the Olympics and I would love to make the team, but it isn’t in my hands.”
She will now await the decision of Team GB selectors, who are due to announce their choices later this month.
Commonwealth backstroke champion Walker-Hebborn will also be sweating on his place after missing his qualification target.
With a medley relay place up for grabs, he should be picked for Rio – with runner-up Liam Tancock also in contention – but believes he needs to improve.
“I’m a bit disappointed with the time,” said Walker-Hebborn, after finishing in 53.73secs.
“I am not one for excuses, but I had a few issues before trials and if I’m firing on all cylinders I know [faster times are] there.”
Davies too missed out on her target time of 59.05secs but was pleased with a 59.64, which put her ahead of rival Lizzie Simmonds.
“I am so pleased,” she told BBC Radio 5 live.
“I was really disappointed with trials last year [which saw her miss the World Championships], but it fuelled me to get more hungry in training.
“I hope it’s enough for Rio.”
Mallett’s first British title was an impressive achievement, but with his winning time four seconds slower than the Rio qualification mark, his best chance of making the Games will come in the 100m butterfly event later in the week.
“Being the British champion was something I wanted to achieve,” said the 26-year-old.
“I want to start making senior international teams and keep moving forward and one day win medals.”
