Rt Revd Tim Dakin has decided to launch the appeals following the two humanitarian crisis'.
The Diocese is calling on local people to support Refugee Action, one of the UK's leading refugee resettlement charities.
The charity is currently working with refugees arriving in the UK by offering them safe shelter, clothing and meals.
The Diocese is also keen to help people affected by the recent flooding in Myanmar.
It is thought hundreds of thousands of people have been impacted by the flooding as well as the resulting landslides.
Families have lost their homes, their possessions and some have been forced to move into camps to live.
Anglican Relief and Development Fund is pairing up with the Diocese of Winchester for this project and wants to help Christians that are worse off to "show the love of Christ to those in need in their own communities."
"I am sure that, like me, people across the Diocese have been shocked by the images they have seen in the news of people forced to flee their homes in Syria," said Rt Revd Tim Dakin.
He added: "Now I am calling on people in this Diocese to support the excellent work being done by Refugee Action to help Syrian refugees who have arrived in the UK.
"This Diocese also has close links with the church in Myanmar, one of our Anglican Communion link partners.
"Flooding in Myanmar in recent months has caused an urgent humanitarian crisis.
"I am therefore also calling for donations to the Anglican Relief and Development Fund to help support the very important work it is doing in supporting those affected by the floods.
"Jesus's parable of the Good Samaritan teaches us to look beyond our immediate horizons to find and help our neighbours in need.
"This is an opportunity to help those who are fleeing persecution and natural disaster.
"I hope everyone will consider donating prayerfully, generously and sacrificially to help these people who are in such desperate need.
"I also invite everyone to join me in praying for those who have been forced to flee their homes."
The Diocese is setting up a special section on its website so that donations can be made online.
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