The BLS has data on job openings. They have a whole section dedicated to job openings and turnover. They have stats on the number of jobs open, hirings, layoffs, and number of people who quit.
Hiring and Separations vs Unemployment
If there are more layoffs then hirings then unemployment will go up and the opposite is true. If you start with an unemployment rate of 10% and the hirings are 1% and separations 2% then you'll expect you'll end up with unemployment of 11%.
Jobs vs Unemployment
Reasonably we should expect that if the job openings are higher then unemployment will be lower and vice versa. But there are more details to it of course. If job openings don't match the skills and experience of the people currently unemployed then the unemployment won't drop quickly. e.g. if theres 1 million jobs open for engineers and 5 million unemployed carpenters then those job openings won't quickly be filled by the job seeking carpenters. Of course the disparity between job openings and job seekers is never that stark.
I'm going to look at the figures as represented in a rate % similar to how unemployment % rate are given. It makes a little more sense to look at a % since the total population generally increases over time.
First of all lets look back over 10 years. Here are the trends in the jobs, hirings, turnover and unemployment rates from 2000 to June 2010:
You can clearly see how the jobs (dark blue) and the unemployment (cyan) show opposite trends. When unemployment goes up, jobs goes down. The trends between hirings and turnover are less obvious. However we can see that when turnover is above hiring that the unemployment trends upwards and vice versa. Look at the section from 2007 to 2009 and you can seed that turnovers are above hiring and that is when unemployment climbed. Back from 2003 to 2005 the hiring is above turnover and unemployment was trending down.
Lets take a closer look at the more recent trends. Here are the rates for the past 2 years from June 2008 to June 2010:
I'm not going to make any predictions on the future. Theres nothing in this data that tells us what direction any of the figures will go from here.