Boris Johnson discussed boosting trade links with the leaders of Japan, Australia and Egypt on the final day of his first international summit.
The Prime Minister also talked about the crisis in Hong Kong in his meeting with Mr Morrison.
Mr Morrison said: "You've just got to get round the table and work it through. I'm not going to create any arbitrary deadlines on this."
He suggested a move to "lock in what we can lock in and get that moving, and I think that will create momentum for the UK for arrangements with Australia and others to be concluded quickly".
"But, at the end of the day, it's got to be in our interests," Mr Morrison said.
"We are not going to sign up to something we don't think helps us and neither will they."
Pushed on whether it could be done in a year, he said: "I would hope so."
Mr Johnson and Japanese counterpart Shinzo Abe committed to pursuing an "ambitious free trade agreement" after Brexit, building on the existing deal with the EU, during their own meeting in Biarritz.
In his meeting with Mr Morrison "the leaders discussed their enthusiasm for an enhanced and deep trading relationship once the UK leaves the EU", Number 10 said.
A Downing Street spokeswoman added: "The two leaders were concerned about the current situation in Hong Kong, and agreed that it was paramount for all sides to remain calm and peaceful and to maintain Hong Kong's high degree of autonomy as guaranteed by the legally binding joint declaration."
Mr Johnson's meeting with Egyptian president Abdel Fattah el-Sisi focused on their "shared desire to strengthen economic, trade and security cooperation", Number 10 said.
The Prime Minister was also meeting outgoing Italian premier Giuseppe Conte today.
In the main business on the agenda, the leaders of the G7 nations considered measures on protecting the environment, although Donald Trump's chair was conspicuously empty at the start of the session.
The US president is a climate change sceptic who once had claimed it is a hoax that was invented by the Chinese.
Mr Johnson used the session to highlight a pledge of £10 million to help prevent the destruction of the Amazon.
As the G7 summit drew to a close, former prime minister Gordon Brown said the club made up of some of the world's richest democracies was "impotent" because of its divisions.
Ahead of the summit, host Emmanuel Macron said there would not be a final communique, an agreed statement backed by all the leaders.
The 2018 G7 summit ended in farce as Mr Trump withdrew his support for the text and Mr Macron was keen to avoid a repeat performance.
Mr Brown told BBC Radio 4's Today: "When you have got an organisation that cannot agree on a communique, that has got no agreed agenda, that's got no agreement even on membership, and has broken down, as far as I can see, over the weekend into small huddles of individuals doing bilateral discussions - you've really got a leaderless world.
"It is rightly called by some the G Zero because the world seems to be more divided than I can remember."