Tory MPs are calling for the Prime Minister to sack her closest advisers after a disastrous election result that ended with the Conservative party losing their majority.
The prime minster had called snap elections in order to bolster her thin majority prior to Brexit negotiations.
Theresa May is expected to finalise her team of ministers later as she seeks to lead a government with the support of the Democratic Unionist Party.
Announcing his resignation on the Conservative Home website, Mr Timothy apologised for his part in the Tories' defeat, which included drafting the unpopular manifesto.
"In particular, I regret the decision not to include in the manifesto a ceiling as well as a floor in our proposal to help meet the increasing cost of social care", he wrote.
Ministers have reportedly accused the two special advisers of acting like deputy Prime Ministers over their stranglehold on power in Whitehall.
Philip Hammond will stay as chancellor, Boris Johnson will remain as foreign secretary and Amber Rudd - narrowly re-elected as an MP after a recount - will continue as home secretary.
Katie Perrior said 10 Downing Street operated in a "pretty dysfunctional" manner during the 10 months in which she acted as Mrs May's director of communications.
She had limited room for manoeuvre after her campaign saw the Tories shed seats and fall eight MPs short of a Commons majority.
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Identifying a problem is half the battle won and now they have to rectify their mistakes and put in a better effort in the field. The tendency will be there, because we all live for these kinds of moments. "We played in Birmingham before".
At the same time, the pressure continues to grow on the prime minister, with nearly two-thirds of Conservative Party members believing that she should resign, according to a survey conducted by the Conservative Home website.
"I was fairly straightforward with her and I told her that there were a number of things that count to me more than party".
Timothy's colleague as joint chief if staff, Fiona Hill, released a brief statement following her resignation from her role at 10 Downing Street.
Writing in the "Times", she said: "What I could never work out was whether May condoned their behaviour and turned a blind eye or didn t understand how destructive they both were".
The deal sits uneasily with some Conservatives because of the DUP's opposition to abortion and same-sex marriage.
Scottish Conservative leader Ruth Davidson. German Chancellor Angela Merkel said she saw no reason to grant the United Kingdom a grace period.
Mrs May refused to quit despite calls from her opponents as members of her own party criticised her campaign and suggested she should consider her position.
Owen Paterson said Theresa May was right "despite all the disappointments" to try to find a way to govern with the support of the Democratic Unionist Party given that the Tories won the most seats and most votes.
But senior figures also cautioned against an immediate leadership election, as the government prepares to start talks on leaving the European Union around June 19.
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