Since a few people have asked for it, here are a few quotes 'n' sources to start learning about early American leaders' beliefs of America as the New Israel.
"We shall be as a City upon a Hill, the eyes of all people are upon us...," Puritan John Winthrop wrote. The Puritans who disembarked in Massachusetts in 1620 believed they were establishing the New Israel. Indeed, the whole colonial enterprise was believed to have been guided by God. They had seen Haley's comet, and believed that it was a sign for a new Millennium or new Order, which they were going to build on the new continent. See "Mayflower" by Nathaniel Philbrick.
"We cannot but acknowledge that God hath graciously patronized our cause and taken us under his special care, as he did his ancient covenant people," Samuel Langdon preached at Concord, New Hampshire in 1788.
George Washington was the "American Joshua," and "Never was the possession of arms used with more glory, or in a better cause, since the days of Joshua, the son of Nun," Ezra Stiles urged in Connecticut in 1783.
Benjamin Franklin and Thomas Jefferson proposed Israelitish images for America's Great Seal. Franklin proposed Moses dividing the Red Sea with Pharaoh's army being overwhelmed by the closing waters. Jefferson urged a representation of the Israelites being led in the wilderness by the pillar of fire by night and the cloud by day.
These are some specific examples. There is more written about it in Russell Kirk's masterpiece "The Roots of American Order".
The Covenant was restored in modernity in America, on a continent that had been kept secret from the rest of the world for generations in order to be a free land expressly for the purpose of re-establishing the Covenant.
And don't forget to read about a New Jerusalem that will be built in America, which also draws parallels between America and Israel. Not that I have seen any early American political leaders write about this topic specifically, but I have seen in John Adams' writings that they knew they were engaged in something amazingly grand with the help of God.