Sure, yes. The first party system was starting to form in 1792–93, during his first term as president. He was very cognizant of this. Thomas Jefferson and Alexander Hamilton, two leaders of opposite parties, were famously rival forces within Washington's cabinet.

Washington professed to despise parties or factionalism. He wrote to Jefferson in 1792:

"How unfortunate, and how much is it to be regretted then, that whilst we are encompassed on all sides with avowed enemies and insidious friends, that internal dissentions should be harrowing and tearing our vitals. The last, to me, is the most serious—the most alarming—and the most afflicting of the two. And without more charity for the opinions and acts of one another in Governmental matters, or some more infalible criterion by which the truth of speculative opinions, before they have undergone the test of experience, are to be forejudged than has yet fallen to the lot of fallibility, I believe it will be difficult, if not impracticable, to manage the Reins of Government or to keep the parts of it together: for if, instead of laying our shoulders to the machine after measures are decided on, one pulls this way and another that, before the utility of the thing is fairly tried, it must inevitably be torn asunder—And, in my opinion the fairest prospect of happiness and prosperity that ever was presented to man, will be lost—perhaps for ever!"

My earnest wish, and my fondest hope therefore is, that instead of wounding suspicions, and irritable charges, there may be liberal allowances—mutual forbearances-and temporising yieldings on all sides. Under the exercise of these, matters will go on smoothly, and, if possible, more prosperously. Without them every thing must rub, the wheels of Government will clog—our enemies will triumph—and by throwing their weight into the disaffected Scale, may accomplish the Ruin of the goodly fabric we have been erecting.

Three days later, Washington wrote a very similar letter to Hamilton that echoed some of the sentiments and language of the letter to Jefferson:

How unfortunate would it be, if a fabric so goodly—erected under so many Providential circumstances—and in its first stages, having acquired such respectibility, should, from diversity of Sentiments, or internal obstructions to some of the acts of Government (for I cannot prevail on myself to believe that these measures are, as yet, the deliberate acts of a determined party) should be harrowing our vitals in such a manner as to have brought us to the verge of dissolution. Melancholy thought! But one, at the same time that it shows the consequences of diversified opinions, when pushed with too much tenacity; exhibits evidence also of the necessity of accommodation; and of the propriety of adopting such healing measures as will restore harmony to the discordant members of the Union, & the governing powers of it.

By 1795, as partisanship grew increasingly intense, Washington again lamented parties in a letter to Timothy Pickering.

But (much indeed to be regretted!) party disputes are now carried to that length, and truth is so enveloped in mist, and false representation that it is extremely difficult to know through what channel to seek it. This difficulty to one, who is of no party, & whose sole wish is to pursue, with undeviating steps a path which would lead this Country to respectability, wealth & happiness is exceedingly to be lamented.

Washington may have disliked the idea of factionalism, but in most ways he governed as a Federalist. In fact, most historians consider him a Federalist. During the 1790s, there were no formal party mechanisms. People didn't register as Federalists. Instead, belonging in a "party" was defined by behavior and identity.

To be sure, Washington resisted the label of being a "Federalist." Indeed, many Federalists of the time attempted to describe their rivals, the Republicans, as a self-interested party while claiming that they alone represented the true interest of society. In this schema, there was one group (Federalists) acting in the interests of the nation, and a party (the Republicans) acting out of self-interest. Through this rhetorical move, they could claim to be opposed to partisanship even as they acted very much like partisans. I don't know if Washington ever used this same rhetoric. It doesn't really sound like him. But I wouldn't be surprised if he was somewhat sympathetic to the ideas.

Is it ever mentioned in the Federalist papers?