Is there any REITs (Real Estate Investment Trust)...?

allow me to invest as small as $20K to start off? If so what company?

Lincoln what?

Yes, lots. I would not put all 20k into one REIT--too much risk. Depending on what your total investment portfolio looks like it might be a bad idea to put all 20k into REITs.

REITs come in quit a few different flavors, just like ice cream. There are apartment REITs--EQR is a very large one. Pays a dividend of about 5.1% and sells at about 36-37 a share. Another is CPT about 1/3 the size of EQR. Pays a dividend of about 5.8% and sells at about 48-49 a share.

Office building REITs--OFC, DRE, and CLI are a few of those. Dividends from 4.3% to 8.4%

Shopping center REITs--BDN is one I think. Dividend above 9%. One of its largest customers is bankrupt though.

Health care REITs--NHI, HCP are a couple

There are several other varieties also including hotel REITs, and storage building REITs. The storage building REITs might be really in demand of late with all the forclosures. Check out PSA

Lincoln is a good one.

Are you talking about traded or untraded REITs? Yes there are several untraded REITS that I know of, Wells REIT is one. Traded REITS are bought and sold like stocks with the prices per share varying like the stocks in the stock market.
You might want to check out the National Association of Real Estate Investment Trusts to learn about this type of investment and what different types of REITs are available.

if u invest thru a discount brokerage u can invest ant amt in a reit

As far as I know, REIT's are sold in shares like ETF's and mutual funds.

I am coming into $40,000 in about one month, and I am considering allocating about $10,000 of it to a REIT. I am not sure yet though.

My favorite website has a REIT section, and it has a list of some REIT's.

Good Luck