Perhaps in a bid to reverse American Meryl Streep playing Brit Margaret Thatcher, Carey Mulligan is now the "top choice" beating out the aforementioned folks and even Emma Stone for the part of Hillary Clinton. This is all super early, but she'll be meeting soon with director James Ponsoldt, and THR says "both sides are eager for it to work." But as always, nothing is final, and other names are still in the atmosphere, so who knows if this'll pan out.
The Black List script by Young Il Kim focuses on Clinton's student days and her rise into the world of politics, her work on the committee to impeach Richard Nixon, and yes, her dalliance with a young sax player named Bill. It seems that everyone involved wants to get this made soon (perhaps because Clinton could run for President in 2016, talk about timing) so we'd wager it's only a matter of time until a decision is made.