"D'vorim Hayyotz'im min Halleiv Nichnasim el Halleiv"

I saw the following quote on the Internet:

"If words come out of the heart, they will enter the heart."
~ Rumi ~

Then it hit me - did Rumi get this idea from us Jews or vice versa?

So I conducted a search for the earliest source for:

"D'vorim Hayyotz'im min Halleiv Nichnasim el Halleiv"

One kind colleague, Rabbi Yehoshua Steinberg, responded as follows:

«It's often quoted in the name of Chazal, but the closest Chazal is Brachot 6a "kol adam sheyesh bo yirat shamayim devorav nishmaim."

Much closer though is 
Sefer haYashar, R"T (13:173): "kol davar sheyatza min halev yikanes balev."»

And R Steinberg add the following T'fillah;

«May our prayers on Rosh Hashanah be from the heart and enter the RBSH"O's heart kivayachol.» 

I Googled the Dates of Birth
For our competing authors -

Rumi's DoB:
September 30, 1207,
Vakhsh, Tajikistan
Jalal ad-Din Muhammad Rumi,

Rabbeinu Tam's DoB:
1100, Ramerupt, France
Rabbeinu Tam

Thus, Rabbeinu Tam preceded Rumi by about a century, yet the precise origin may remain shoruded in mystery, covered by the sands of time.

Best Regards,
RRW