Joko Widodo has a private meeting with the British Prime Minister on Tuesday, as part of a tour of Europe which started last month.
Amnesty wants David Cameron to raise the case of a Christian woman who was caned 30 times in front of hundreds of people earlier this month in Aceh province, after being convicted of selling alcohol.
Indonesia is the most populous Muslim country in the world with around 9 in 10 Indonesians practising Islam.
In Aceh province Sharia (Islamic) law is in force, which forbids the production or consumption of alcohol.
Amnesty is also urging Mr Cameron to highlight the case of British woman Lindsay Sandiford, who is on death row in Indonesia.
The 59-year-old former legal secretary from Cheltenham was convicted of drug smuggling offences and sentenced to death in 2013.
Papang Hidayat, Indonesia researcher at Amnesty International, said: "Trade deals must not be made by the UK government at the expense of condemning the deeply troubling human rights abuses which have taken place under president Widodo's leadership.
"This is a president whose ruthless 'war on drugs' led to 14 executions last year to the horror of the watching world, with more promised for this year.
"Mr Cameron should also be raising alarm over more than 100 canings from last year for perceived moral offences, including a 60-year-old Christian woman who allegedly sold some alcohol in Aceh province."
